Straight Lines Questions for CBSE Class 11th

The number of values of a for which the lines 2x + y – 1 = 0, ax + 3Y – 3 = 0, and 3x + 2Y – 2 = 0 are concurrent is

To remove the first degree terms in the equation 4 x 2 − 9 y 2 − 8 x + 36 y + 4 = 0 the origin in shifted to the point

The equations to a pair of opposite sides of a parallelogram are x 2 − 5 x + 6 = 0 and y 2 − 6 y + 5 = 0 . The equations to its diagonals, are

T h e a n g l e b e t w e e n t h e p a i r o f l i n e s w h o s e c o m b i n e d e q u a t i o n i s 2 x 2 + 5 x y + 3 y 2 + 6 x + 7 y + 4 = 0 i s cos – 1 k 26 t h e n k =

If x , y ∈ R and 15 x + 8 y = 34 , then x 2 + y 2 min , is equal to

If two vertices of a triangle are (-2, 3) and (5, -l), the orthocenter lies at the origin, and the centroid on the line x + y = 7 , then the third vertex lies at

The distance between the lines given by ( x + 7 y ) 2 − 42 = 4 2 ( x + 7 y ) is

The equation of straight line passing through (-a, 0) and making a triangle with the axes of area T is

If A x 1 , y 1 , B x 2 , y 2 , C x 3 , y 3 are the vertices of a triangle, then the equation x y 1 x 1 y 1 1 x 2 y 2 1 + x y 1 x 1 y 1 1 x 3 y 3 1 =0 represents

A straight line L through the point ( 3 , − 2 ) is inclined at an angle 60 ∘ to the line 3 x + y = 1 . If L also intersects the x -axis, then the equation of L is

The locus of point of intersection of the line y + m x = a 2 m 2 + b 2 and m y − x = a 2 + b 2 m 2 is

If the line passing through P(1, 2) making an angle π 4 with the x-axis in the positive direction meets the pair of lines x 2 + 4 x y + y 2 at A and B , then P A ⋅ P B =

One of the diameter of the circle circumscribing the rectangle ABCD is 4 y = x + 7 . If A ( − 3 , 4 ) , B ( 5 , 4 ) . Then the equation of the side D C ¯ is

P and Q are points on theline joining A ( − 2 , 5 )    and   B ( 3 , 1 ) such that AP = PQ = QB . Then, the distance of teh midpoint of PQ from the origin is

The image of the point (3, –8) under the transformation (x, y) (2x + y, 3x – y) is

Consider the point A ≡ (0, 1) and B ≡ (2, 0). Let P be a point on the line 4x + 3y + 9 = 0. Coordinates of the point P such that |PA – PB| is maximum, are

lf points a 2 , 0 , 0 , b 2 and ( 1 , 1 ) are collinear, then

If O is the origin and P x 1 , y 1 , Q x 2 , y 2 are two points, then O P . O Q cos ⁡ ∠ P O Q

The equation of the lines through the point 2 , 3 and making an intercept of length 2 units between the lines y + 2 x = 3   and   y + 2 x = 5 is

T h e r a t i o t h a t x – a x i s d i v i d e s t h e l i n e s e g m e n t j o i n i n g t h e p o i n t s ( 2 , – 3 ) a n d ( 5 , 6 ) i s ( i n t e r n a l l y )

A rectangle A B C D , where A ≡ ( 0 , 0 ) , B ≡ ( 4 , 0 ) , C ≡ ( 4 , 2 ) , D ≡ ( 0 , 2 ) , undergoes the following transformations successively: i. f 1 ( x , y ) ( y , x ) ii. f 2 ( x , y ) ( x + 3 y , y ) iii. f 3 ( x , y ) ( ( x − y ) / 2 , ( x + y ) / 2 ) The final figure will be

If the x-intercept of the line y = mx + 2 is greater than 1 2 , then the gradient of the line lies in the interval

The point (4, I ) undergoes the following three transformations successively. i . Reflection about the line y: x. ii. Translation through a distance 2 units along the positive direction of the x-axis. iii. Rotation through an angle π 4 about the origin in the counterclockwise direction. Then the final position of the point is given by the co ordinates

The straight line whose sum of the intercepts on the axes is equal to half of the product of the intercepts, passes through the point

T h e p o i n t s ( – a , – b ) , ( a , b ) , ( a 2 , a b ) f o r m s

I f p i s t h e l e n g t h o f t h e p e r p e n d i c u l a r f r o m t h e o r i g i n t o t h e l i n e x a + y b = 1 t h e n

A line with positive rational slope passes through the point A(6, 0) and is at a distance of 5 units from B(1, 3). If the slope of the line is m, then the value of 75m is

A variable line ‘ L ‘ is drawn through O ( 0 , 0 ) to meet the lines L 1 : y − x − 10 = 0 and L 2 : y − x − 20 = 0 at the points A and B respectively. A point P is taken on ‘ L ′ such that 2 O P = 1 O A + 1 O B . Locus of ‘ P ′ is

If AD, BE and CF are the altitudes of a triangle ABC whose vertex A is the point (-4, 5). The coordinates of the points E and F are (4,1) and (-1, -4) respectively, then equation of BC is

If the slope of one of the lines represented by a x 2 + 2 h x y + b y 2 = 0 is the square of the other, then a + b h + 8 h 2 a b =

The number of triangles which are obtuse and which have the points (8,9), (8, 16) and (20, 25) as the feet of perpendiculars drawn from the vertices on the opposite sides is

Locus of the points which are at equal distance from 3x + 4y – 11 = 0 and 12x + 5y + 2 = 0 and which is near the origin is

If AD, BE and CF are the altitudes of ∆ ABC whose vertex A is the point (-4, 5). Also, the coordinates of the points E and F are (4,1) and (-1, -4) respectively. Then, equation of BC is

Given that A ≡ ( 1 , − 1 ) and locus of B is x 2 + y 2 = 16 . If P divides A B in the ratio 3 : 2 then locus of P is

The locus of image of the point λ 2 , 2 λ in the line mirror x − y + 1 = 0 where λ is a parameter is

A line is passing through the points A 2 , 3 , B 2 , 7 . The point where the line perpendicular to the line A B and passing through 1 , 4 meets the line A B is

Let 3 , 4 are the intercepts of a line then intercepts of a line which is perpendicular to the line and passing through the point 1 , 3 is

The perpendicular distance from the point 1 , 1 to the line which is perpendicular to 4 x − 3 y + 1 = 0 and passing through 3 , 5 is

The quadrilateral formed by the lines whose equations are given by x 2 − 5 x + 6 = 0 and y 2 − 14 y + 40 = 0 is

If the line 4 x − 5 y + 41 = 0 is the perpendicular bisector of the line segment joining the points A ( 1 , 1 ) and B . The image of B in x -axis is ( a , b ) then the numerical value of a+b is

Reflection of the point 1 , 1 with respect to the line 4 x + 3 y − 5 = 0 is α , β then α + β

The perpendicular distance from origin to the line 4 x + 3 y − 5 = 0 is

Equation of line passing through the point 5 , 7 and slope is equal to slope of the line joining through the points 3 , 7 and 5 , 3 is a x + b y + c = 0 then a + b =

If a , b are intercepts of the line which is perpendicular to the line passing through the points 3 , 5 , 4 , 3 and passing through the point 5 , 6 then a b =

A stralght line passing through P ( 3 , 1 ) meets the coordinate axes at A and B . It is given that the distance of this straight line from the origin O is maximum. The area of triangle OAB is equal to

If A(0,0),B(1,0) and C ( 1 2 , 3 2 ) are vertices of a triangle, then the centre of the circle for which the lines AB,BC and CA re tangents is

In ΔABC , if the orthocenter is (1,2) and the circumcenter is (0,0) then centroid of ΔABC is

The point of intersection of lines 2 x + 3 y + 4 = 0 and 6 x − y + 12 = 0 is

The slope of the line which is perpendicular to the angular bisector of axes which makes obtuse angle is

If the lines y = 2 x + 3 , y = 3 x + 4 and y = mx + 7 are concurrent then m=

The angle between the pair of lines x 2 − 2 xysecα + y 2 = 0 is

The equation ax 2 + 2 hxy + by 2 + 2 gx + 2 fy + c = 0 represents a pair of parallel lines then ab =

If ax 2 + 2 hxy + by 2 + 2 gx + 2 fy + c = 0 represents a pair of parallel lines then b a

The equation to the pair of lines passing through origin and perpendicular to 5 x 2 + 7 xy + y 2 = 0

If x 2 + 2 xy + ky 2 + 4 x − 2 y + l = 0 represents pair of perpendicular lines then

The circum centre of the triangle whose sides are 3 x − y − 5 = 0 , x + 2 y − 4 = 0 , 5 x + 3 y + 1 = 0 is α , β then α 2 + β 2 =

The number of lines that can be drawn through the point P at a distance of d meters from the point Q if PQ > d are

The orthocentre of triangle formed by points 0 , 0 3 , 4 2 , 3 is − k , k then k is

If the slope of one line of the pair of lines represented by ax 2 + 10 xy + y 2 = 0 is four times the slope of the other, then a is equal to

The triangle formed by the pair of lines 11 x 2 + 16 xy − y 2 = 0 and x − 2 y + 13 = 0 is

If a straight line cuts intercepts from the axes of coordinates the sum of the reciprocals of which is a constant k, then the line passes through the fixed point

P is a point on either of the two lines y – 3 x = 2 at a distance of 5 units from their point of intersection. The coordinates of the foot of the perpendicular from P on the bisector of the angle between them are

A line is drawn from the point P α , β making an angle θ with the positive direction of x-axis, to meet the line ax + by + c = 0 at Q. The length of PQ is

If the line through the points (h, 7) and (2, 3) intersects the line 3x – 4y – 5 = 0 at right angles, then the value of h is

The equation a x 2 + b y 2 + c x + c y = 0 represents a pair of straight lines if

If one of the lines given by 2 c x 2 + 2 x y − c 2 − 1 y 2 = 0 is 2 x + 3 y = 0 then the integral value of c is

The orthocentre of the triangle formed by the lines x y = 0 and 2 x + 3 y − 5 = 0 is

The equation a x 2 + 2 h x y + a y 2 = 0 represent a pair of coincident lines through the origin if

The point a 2 , a lies between the straight lines x + y = 6 and x + y = 2 for

If the line passing through ( 3 , 4 ) and ( x , 5 ) makes 135 ° angle with the positive direction of x – axis, then x =

Triangle formed by x 2 − 3 y 2 = 0 and x = 4 is

The number of points equidistant to three given distinct non-collinear points, is

If the axes are rotated through an angle of 30 ° in clockwise direction, the point ( 4 , 2 3 ) in the new system is

The area of the parallelogram formed by the lines 2 x − 3 y + a = 0 , 3 x − 2 y − a = 0 , 2 x − 3 y + 3 a = 0 and 3 x − 2 y − 2 a = 0 in square unit is

Let A ( a , b ) be a fixed point and O be the origin of coordinates, If A 1 is the mid-point of O A , A 2 is the mid-point of -. A 1 , A 3 is the mid-point of A A 2 and so on. Then the coordinates of A 1 are

A ( − 5 , 0 ) and B ( 3 , 0 ) are two of the vertices of a triangle A B C . Its area is 20 square ems. The vertex C lies on the line x − y = 2 . The coordinates of C , are

If A ( cos ⁡ α , sin ⁡ α ) , B ( sin ⁡ α , − cos ⁡ α ) , C ( 1 , 2 ) are the varties △ A B C then as a varies the locus of its centroid, is

The value of k for which the lines 2 x − 3 y + k = 0 3 x − 4 y − 13 = 0 and 8 x − 11 y − 33 = 0 are concurrent, is

The medians A D and B E of the triangle with vertices A ( 0 , b ) , B ( 0 , 0 ) and C ( a , 0 ) are mutually perpendicular, if

A straight line through point A ( 3 , 4 ) is such that its intercept between the axes is bisected at A . Its equation is

Let the perpendiculars from any point on the line 2 x + 11 y = 5 upon the lines 24 x + 7 y − 20 = 0 and 4 x − 3 y − 2 = 0 have the lengths p 1 and p 2 respectively. then, p 1 – p 2 is equal to

The x-coordinates of the incentre of the triangle where the mid-points of the sides are ( 0 , 1 ) , ( 1 , 1 ) and ( 1 , 0 ) is

If the centroid of the triangle having its vertices ,(1, a), (2, b) and (c 2 ,- 3) lies on X-axis, then

If one of the lines given by 6 x 2 − x y + 4 c y 2 = 0 is 3 x + 4 y = 0 , then c equals

The incentre of the triangle having its vertices at O (0, 0), A(5, 0) and B (0, 12) is

The line x a − y b = 1 cuts the x – axis at P . The equation of the line through P perpendicular to the given line, is

The vertices of a triangle are at O (0, 0), A (a, 0) and B (0, b). The distance between its circumcentre and orthocentre is

If P ( 1 , 0 ) , Q ( − 1 , 0 ) and R ( 2 , 0 ) are three given points, then the locus of point S satisfying the relation S Q 2 + S R 2 = 2 S P 2 , is

If the point x 1 + t x 2 − x 1 , y 1 + t y 2 − y 1 ) divides the join of x 1 , y 1 and x 2 , y 2 internally, then

The equation of the straight line which is perpendicular y = x and passes through (3, 2) will be given by

If a vertex of an equilateral triangle is the origin and the side opposite to it has the equation x + y = 1 , then the orihocenire of the triangle is

If a line joining two points A ( 2 , 0 ) and B ( 3 , 1 ) is rotated about A in anticlockwise direction through an angle 15 ° such that the point B goes to C in the new position, then the coordinates of C are

The coordinates of the point where origin to be shifted so that the equation y 2 + 4 y + 8 x − 2 = 0 will not contain the term in y and the constant term, are

Let two points be A ( 1 , – 1 ) and B ( 0 , 2 ) . If a point P x ‘ , y ‘ be such that the area of Δ PAB = 5 sq units and it lies on the line 3 x + y – 4 λ = 0 , then a value of λ is :

If P ( 1 + ( α / 2 ) , 2 + ( α / 2 ) ) be any point on a line, then the range of values of α for which the point P lies between the parallel lines x + 2 y = 1 and 2 x + 4 y = 15 is

If x c + y d = 1 be any line through the intersection of lines x a + y b = 1 and x b + y a = 1 , then

The coordinates of a point P on the line 2 x – y + 5 = 0 such that | P A – P B | is maximum where A is ( 4 ¯ – 2 ) and B is ( 2 , – 4 ) will be:

Suppose A , B are two points on 2 x − y + 3 = 0 and P ( 1 , 2 ) is such that P A = P B , then the mid-point of A B is

T h e l e n g t h o f t h e a l t i t u d e t h r o g u h A o f t h e t r i a n g l e A B C   w h e r e A ( – 3 , 0 ) , B ( 4 , – 1 ) , C ( 5 , 2 ) i s 22 k t h e n k =

Let 0 < α < π / 2 be a fixed angle. If P ≡ ( cos ⁡ θ , sin ⁡ θ ) and Q ≡ ( cos ⁡ ( α − θ ) , sin ⁡ ( α − θ ) ) , then Q is obtained from P b y t h e

The range of value of α such that ( 0 , α ) lies on or inside the triangle formed by the lines y + 3 x + 2 = 0 , 3 y − 2 x − 5 = 0 , 4 y + x − 14 = 0 is

Let O(0, 0), P(3, 4), and Q(6,0) be the vertices of triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are

T h e o r t h o c e n t r e o f t h e t r i a g l e f o r m e d b y t h e v e r t i c e s ( 0 , 0 ) , ( 0 , 3 ) , ( 4 , 0 ) i s

ln ∆ ABC, if the orthocenter is (1, 2) and the circumcenter is (0, 0), then centroid of ∆ ABC is

If the pair of lines 6 x 2 − α x y − 3 y 2 − 24 x + 3 y + b = 0 intersect on x -axis, then α is equal to

T h e e q u a t i o n o f a c u r v e r e f e r r e d t o t h e n e w a x e s w h e n t h e o r i g i n s h i f t e d t o ( 4 , 5 ) w i t h o u t c h a n g i n g t h e d i r e c t i o n o f a x e s i s X 2 + Y 2 = 36 t h e n t h e e q u a t i o n o f t h e c u r v e w i t h r e f e r e n c e t o t h e o r i g i n a l a x e s

A line L is drawn from P ( 4 , 3 ) to meet the lines L 1 and L 2 given by 3 x + 4 y + 5 = 0 and 3 x + 4 y + 15 = 0 at points A and B , respectively. From A ¯ , a line perpendicular to L is drawn meeting the line L 2 at A 1 . Similarly, from point B , a line perpendicular to L is drawn meeting the line L 1 at B 1 . Thus, a parallelogram A A 1 B B 1 is formed. Then the the equation of L so that the area of the parallelogram A A 1 B B 1 is the least is

I f t h e c i r c u m c e n t r e o f a n a c u t e a n g l e d t r i a n g l e l i e s a t t h e o r i g i n a n d t h e c e n t r o i d i s t h e m i d d l e p o i n t o f t h e l i n e j o i n i n g t h e p o i n t s a 2 + 1 , a 2 + 1 a n d 2 a , – 2 a t h e n t h e o r t h o c e n t r e i s

A straight line L with negative slope passes through the points (8, 2) and cuts the positive coordinate axes at points P and Q. As L varies then the absolute minimum value of OP + OQ is (O is origin)

If the line x a + y b = 1 moves in such a way that 1 a 2 + 1 b 2 = 1 c 2 where c is a constant, then the locus of the foot of perpendicular from the origin on the straight line is

T w o p a i r s o f l i n e s 2 x 2 + 6 x y + y 2 = 0 a n d 4 x 2 + 18 x y + y 2 = 0

The maximum value of y = ( x − 3 ) 2 + x 2 − 2 2 − x 2 + x 2 − 1 2 is

A light ray coming along the line 3x + 4y = 5 gets reflected from the line ax + by = 1 and goes along the line 5x – 12y = 10. Then,

The equations of the diagonals of square formed by lines x =0 , y = 0, x = 1 and y = 1 are

The number of lines drawn from the point (4, -5) so that its distance from (-2, 3) will be equal to 12 is

The equation of a line which is parallel to the line common to the pair of lines given by 6 x 2 − x y − 12 y 2 = 0 and 15 x 2 + 14 x y − 8 y 2 = 0 and at a distance of 7 units from it is

The equation of the bisector of the acute angle between the lines 2x – y + 4 = 0 and x – 2y = 1 is

If ( 0 , 0 ) is orthocentre of triangle formed by A ( cos ⁡ α , sin ⁡ α ) , B ( cos ⁡ β , sin ⁡ β ) and C ( cos ⁡ γ , sin ⁡ γ ) then ∠ B A C is

The line y = 2x + 4 is shifted 2 units along +y axis, keeping parallel to itself and then I unit along +x axis direction in the same manner, then equation of the line in its new position is

The number of integral values of m, for which the x-coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + I is also an integer, is

For a> b> c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than 2 2 . then

Suppose A , B are two points on 2 x − y + 3 = 0 and P ( 1 , 2 ) is such that P A = P B . Then the mid-point of A B is

The straight lines represented by ( y − m x ) 2 = a 2 1 + m 2 and ( y − n x ) 2 = a 2 1 + n 2 form a

Given A ≡ ( 1 , 1 ) and A B is any line through it cutting the x -axis at B . If A C is perpendicular to A B and meets the y -axis in C , then the equation of the locus of midpoint P o f B C i s

In an isosceles triangle O A B , O is the origin and O A = O B = 6 . The equation of the side A B is x − y + 1 = 0 . Then the area of the triangle is

If a vertex of an equilateral triangle is the origin and the side opposite to it has the equation x + y = 1 then the orthocentre of the triangle is

P is a point on the line y + 2x = 1, and Q and R are two points on the line 3y + 6x = 6 such that triangle PQR is an equilateral triangle. The length of the side of the triangle is

The line x 3 + y 4 = 1 meets the y and x-axis at A and B, respectively. A square ABCD is constructed on the line segment AB away from the origin. The coordinates of the vertex of the square farthest from the origin are

Two consecutive sides of a parallelogram are 4 x + 5 y = 0 and 7 x + 2 y = 0 . If the equation of one diagonal is 11 x + 7 y = 9 , then the equation to the other diagonal is

The equation of line belonging to the family of lines p ( 2 x + 3 y − 13 ) + q ( x − y + 1 ) = 0 , and which is at maximum distance from the orioin is

If k = a b a + b then the line x a + y b = 1 passes through the fixed point, then that point is

The angle between the line x + 2 y − 5 = 0 and the positive x – axis is θ then the value of sin θ is

Suppose that the lines L 3 and L 4 are the angular bisectors of L 1 and L 2 . If the angle between L 1 and L 3 is 30 ° then the angle between L 2 and L 4 is

The points ( 2 , 5 ) , ( 2 , 7 ) , ( 1 , 7 ) forms a triangle. The inclination of the line joining the ortho centre and the circum centre of the triangle is

The tangent of angle between the lines whose intercepts on the axes are 4 , − 5 and 5 , − 4 respectively

A line is passing through the points ( 1 , 3 ) and ( 5 , 7 ) , then its inclination is

The image of the centroid of the triangle whose ortho centre is at origin and the circum centre at 3 , 3 in the line which is equally inclined with axes in the first quadrant is

The triangle formed by the lines 2 x + 3 y − 5 = 0 , x − y + 4 = 0 and 6 x − 4 y + 1 = 0 is

The inclination of the line passing through the point of intersection of lines given by y = x and the point 2 , 0 is

A particle p moves from the point A ( 0 , 4 ) to the point ( 10 , − 4 ) . The particle P can travel the upper-half plane { ( x , y ) | y ≥ 0 } at the speed of 1 m/s and the lower-halfplane { ( x , y ) | y ≤ 0 } at the speed of 2 m/s. The coordinates of a point on the x-axis, if the sum of the squares of the travel times of the upper- and lower-half planes is minimum, are

If the point ( x 1 + t ( x 2 − x 1 ) , y 1 + t ( y 2 − y 1 ) ) divides the join of ( x 1 , y 1 ) and ( x 2 , y 2 ) internally, then

In triangle ABC, angle B is right angle and AC=2. If coordinates of A and B are (2,2) and (1,3), respectively, then the lenght of median AD is

ABCD is a rectangle with A ( − 1 , 2 ) , B ( 3 , 7 )   and   AB : BC = 4 : 3 . If P is the centre of the rectangle, the the distance of P from each corner is equal to

Statement 1: If in a triangle, orthocentre, circumcentre and centroid are rational points, then its vertices must also be rational points. Statement 2: If the vertices of a triangle are rational points, then the centroid, circumcentre are also rational points.

Consider three points P ≡ ( − sin ( β − α ) , − cosβ ) , Q ≡ ( cos ( β − α ) , sinβ ) ,    and    R ≡ ( cos ( β − α + θ ) , sin ( β − θ ) ) ,   where   0 < α , β , θ < π 4 .   Then

If the vertices of a triangle are ( 5 , 0 ) , ( 3 , 2 ) ,    and   ( 2 , 1 ) , then the orthocenter of the triangle is

Point of intersection of family of lines given by 3 x + 4 y − 7 + λ 5 x − 2 y − 3 = 0 is

The image of the line 2 x − y − 1 = 0 with respect to the line 3 x − 2 y + 4 = 0 is

The image of the line x − y + 3 = 0 on x-axis is px + qy + r = 0 then the value of p + q + r =

The angle between the pair of lines 5 x 2 + 4 xy − 5 y 2 = 0 is

The acute angle between the pair of lines cosα + sinα x 2 − 2 xycosα + cosα − sinα y 2 = 0 is

If the lines represented by the equations kx 2 + 2 xy + y 2 + 2 x + 2 y + 4 = 0 are parallel to each other then k =

The distance between the pair of parallel lines 2 x 2 + 4 xy + 2 y 2 + 3 x + 3 y + 1 = 0 is

If the equation ax 2 + 2 hxy + by 2 + 2 gx + 2 fy + c = 0 represents a pair of parallel lines b a =

The equation to the pair of lines parallel to 2 x 2 + 4 xy + y 2 = 0 and passing through 1 , 2 is 2 x 2 + 4 xy + y 2 + kx − 8 y + 14 = 0 then k=

A ray of light along x + 3 y = 3 reflected on reaching x-axis the equation of the reflected ray is y = mx + c then c m =

If one of the lines represented by 2 x 2 + 2 hxy + 3 y 2 = 0 be perpendicular to one of the lines given by 3 x 2 − 2 h ‘ xy + 2 y 2 = 0 , then

The orthocentric of the triangle formed by the lines xy + 3 x + 3 y + 9 = 0 and the line 3 x + 4 y − 5 = 0 is

The triangle formed by the pair of lines 2 x 2 + 4 xy + 2 y 2 = 0 and the line x + 2 y + 3 = 0 is

If x 2 − y 2 = 0 , lx + 2 y = 1 form a isosceles triangle, then l =

The point 1 , β lies on or inside the triangle formed by the lines y = x, x-axis and x + y = 8, if

A line passing through the point P(4, 2), meets the x-axis and y-axis at A and B respectively. If O is the origin, then locus of the centre of the circum circle of ∆ OAB is

Through the point (1, 1), a straight line is drawn so as to form with coordinate axes a triangle of area S. The intercepts made by the line on the coordinate axes are the roots of the equation

Without changing the direction of coordinates axes, origin is transferred to ( α , β ) so that the linear terms in the equation x2 + y2 + 2x – 4y + 6 = 0 are eliminated. The point ( α , β ) is

If x 1 , x 2 , x 3 as well as y 1 , y 2 , y 3 are in G.P. with the same common ratio, then the points (x 1 , y 1 ), (x 2 , y 2 ) and (x 3 ,y 3 )

If a, c, b are three terms of a G.P., then the line ax + by + c = 0

The area of the region enclosed by 4 |x| + 5 |y| ≤ 20 is

The diagonals of a parallelogram PQRS are along the lines x + 3y = 4 and 6x – 2y = 7. Then PQRS must be a

If a triangle has its orthocentre at (1, 1) and circumcentre at 3 2 , 3 4  then the coordinates of the centroid of the triangle are

A rectangle has two opposite vertices at the points (1, 2) and (5, 5). If the other vertices lie on the line x = 3, then the coordinates of the other vertices are

The condition to be imposed on β so that (0, β ) lies on or inside the triangle having sides y + 3x + 2 = 0, 3y – 2x – 5 = 0 and 4y + x – 14 = 0 is

If a, b, c form a G.P., then twice the sum of the ordinates of the points of intersection of the line ax + by + c = 0 and the curve x + 2y 2 = 0 is

If a, b, c form a G.P., then twice the sum of the ordinates of the points of intersection of the line ax + by + c = 0 and the curve x + 2y 2 = 0 is

The point (2t 2 + 2t + 4, t 2 + t + 1) lies on the line x + 2y = 1 for

The perpendicular from the origin to a line L meets it at the point ( 3 , – 9 ) , equation of the line L is

The y-axis and the lines ( a 5 – 2 a 3 ) x + ( a + 2 ) y + 3 a = 0 and ( a 5 – 3 a 2 ) x + 4 y + a – 2 = 0 are concurrent for

The equation a x 2 + 2 h x y + b y 2 = 0 represents a pair of perpendicular lines if

A straight line through the point A ( 3 , 4 ) is such that its intercept between the axes is bisected at A . Its equation is

The line joining A ( b cos a , b sin a ) and B ( a cos b , a sin b ) is produced to the point M ( x , y ) so that A M : M B = b : a , , then x cos α + β 2 + y sin α + β 2 =

If one of the diagonals of a square is along he line x = 2 y and one of its vertices is (3, 0), then its side through this vertex nearer to the origin is given by the equation

Let O be the origin, A ( 1 , 0 ) and B ( 0 , 1 ) and P ( x , y ) are points such that x y > 0 and x + y < 1 , then

Perimeter of the quadrilateral bounded by the coordinate axis and the lines x + y = 50 and 3 x + y = 90 is

The locus represented by the equation ( x − y + c ) 2 + ( x + y − c ) 2 = 0 is

Equation of the line equidistant from the parallel lines 9 x + 6 y − 7 = 0 and 3 x + 2 y + 6 = 0 is

If every point on the line ( a 1 – a 2 ) x + ( b 1 – b 2 ) y = c is equidistant from the points ( a 1 , b 1 ) and ( a 2 , b 2 ) then 2 c =

The number of lines that can be drawn through the point (4, – 5) at a distance 12 from the point (– 2, 3) is

A line through P ( 1 , 2 ) is such that it makes unequal intercepts on the axes, and the intercept between the axes is trisected at P, an equation of the line is

If the slope of one of the lines represented by a x 2 + 2 h x y + b y 2 = 0 be the square of the other, then a + b h + 8 h 2 a b is equal to

The straight lines x + y – 4 = 0, 3x + y – 4 = 0 and x + 3y – 4 = 0 form a triangle which is

The incentre of the triangle with vertices ( 1 , 3 ) , ( 0 , 0 ) and ( 2 , 0 ) is

If non zero numbers a, b, c are in H.P, then the straightline x a + y b + 1 c = 0 always passes through a fixed point. That point is

Equation of the line passing through the point ( a – 1 , a + 1 ) and making zero intercept on both axes is

If a vertex of a triangle is ( 1 , 1 ) and the mid points of two sides through this vertex are ( – 1 , 2 ) and ( 3 , 2 ) , then centroid of the triangle is

A rectangle has two opposite vertices at the points (1, 2) and (5, 5). If the other vertices lie on the line x = 3 , , the coordinates of the vertex nearer the axis of x are

The extremities of a diagonal of a parallelogram are the points ( 3 , – 4 ) and ( – 6 , 5 ) . If third vertex is ( – 2 , 1 ) then the coordinates of the fourth vertex are

The distance between the orthocentre and the circumcentre of the triangle with vertices (0, 0),(0, a ) and ( b , 0) is

The slopes of the line which passes through the origin, and the mid-point of the line segment joining the points. P ( 3 , – 4 ) and Q ( – 5 , – 2 ) is

The equations to the sides of a triangle are x – 3 y = 0 , 4 x + 3 y = 5 and 3 x + y = 0 . The line 3 x – 4 y = 0 passes through the

The centroid of a triangle lies at the origin and the coordinates of its two vertices are (– 8, 7) and (9, 4). The area of the triangle is

The straight lines 4 x − 3 y − 5 = 0 , x − 2 y = 0 , 7 x + y − 40 = 0 and x + 3 y + 10 = 0 form

Let A 0 A 1 A 2 A 3 A 4 A 5 be a regular hexagon inscribed in a circle of unit radius then the product of the lengths of the line segments A 0 A 1 , A 0 A 2 and A 0 A 4 is

The straight line x + 2 y − 9 = 0 , 3 x + 5 y − 5 = 0 and a x + b y − 1 = 0 are concurrent if the straight line 35 x − 22 y + 1 = 0 passes through the point

The points P ( a , b ) , Q ( c , d ) , R ( a , d ) and S ( c , b ) , where a , b , c , dare distinct real numbers, are

The inclination of the straight line passing through the point A ( – 3 , 6 ) and the mid-point of the line joining the B ( 4 , − 5 ) and C ( − 2 , 9 ) , with x – axis is

If non-zero numbers a, b, c are in H.P., then the straight line x a + y b + 1 c = 0 always passes through a fixed point. That point is

If a vertex of a triangle is ( 1 , 1 ) and the mid-points two sides through this vertex are ( – 1 , 2 ) and ( 3 , 2 ) , then k centroid of the triangle, is

If the points A ( 1 , 1 ) , B ( − 1 , − 1 ) and C 3 , 3 are the vertices of a triangle, then triangle is

If the coordinates of the mid-points of side AB and of △ A B C are D ( 3 , 5 ) and E ( − 3 , − 3 ) respectively, the BC=

A triangle ABC lying in the first quadrant has two vertices at A ( 1 , 2 ) and B ( 3 , 1 ) . If ∠ B A C = 90 ∘ and area Δ ( A B C ) = 5 5 sq. units then the abscissa of the vertex C is

The area of the quadrilateral whose vertices are ( 1 , 2 ) , ( 6 , 2 ) , ( 5 , 3 ) and ( 3 , 4 ) , is

One possible condition for the three points ( a , b ) , ( b , a ) and a 2 , − b 2 to be collinear is

The area of the triangle with vertices at ( a , b + c ) , ( b , c + a ) and ( c , a + b ) , is

The locus of a point P which divides the line joining ( 1 , 0 ) and ( 2 cos ⁡ θ , 2 sin ⁡ θ ) internally in the ratio 2 : 3 for all 0 , is a

If O is the origin and P x 1 , y 1 , Q x 2 , y 2 are two points O P × O Q sin ⁡ ∠ P O Q =

If the distance of any point ( x, y) from the origin is defined as d (x, y) = max {I x I, I y I} , then the locus of the point (x, y), where d (x, y) =1, is

If each of the vertices of a triangle has integer: coordinates, then the triangle will not be

If the coordinates of the vertices of a triangle an: rational numbers, then which of the following points of the triangle may not have rational coordinates?

The coordinates of two points A and Bare ( 3, 4) and (5, – 2) respectively. If P is a point not lying on any of the coordinate axes such that P A = P B and Area of of ∆ P A B = 10 , then the coordinates of P, are

If t 1 , t 2 and t 3 are distinct, the points t 1 , 2 a t 1 + a t 1 3 t 2 , 2 a t 2 + a t 2 3 and t 3 , 2 a t 3 + a t 3 3 are collinear if

If x 1 , x 2 , x 3 as well as y 1 , y 2 , y 3 are in GP with same common ratio, then the points P x 1 , y 1 , Q x 2 , y 2 and R x 3 , y 3

Orthocentre of the triangle whose sides are given by 4 x − 7 y + 10 = 0 , x + y − 5 = 0 and 7 x + 4 y − 15 = 0 is

Angles made with the x – axis by two lines drawn through the point ( 1 , 2 ) and cutting the line x + y = 4 at a distance 6 3 from the point ( 1 , 2 ) are

The orthocentre of the triangle with vertices 2 , 3 − 1 2 , 1 2 , − 1 2 and 2 , − 1 2 , is

The equation of the bisector of that angle between the lines x + y = 3 and 2 x – y = 2 which contains the point (1, 1) is

The equation of the line AB is y = x. If A and B lie on the same side of the line mirror 2x – y = 1, then the equation of the image of AB, is

If the circumcentre of a triangle lies at the origin and the centroid is the middle point of the line joining the point is a 2 + 1 , a 2 + 1 and ( 2 a , − 2 a ) then the orthocentre satisfies the equation

The equation of the straight line which passes through the point (1, – 2) and cuts off equal intercepts from the axes will be

If the centroid and circumcentre of a triangle are ( 3,3) and ( 6, 2) respectively, then the orthocenire, is

If α , β , γ are the real roots of the equation x 2 − 3 a x 2 + 3 b x − 1 = 0 then the centroid of the triangle with metrices α , 1 α , β , 1 β and γ , 1 γ is at the point

The orthocentre of the triangle formed by the lines x = 2 , y = 3 and 3 x + 2 y = 6 is at the point

The distance between the parallel lines y = 2 x + 4 and 6 x = 3 y + 5 , is

The bisector of the acute angle formed between the lines 4 x – 3 y + 7 = 0 and 3 x – 4 y + 14 = 0 has the equation

If one of the diagonals of a square is along the line x = 2 y and one of its vertices is (3, 0), then its sides through this vertex are given by the equations

Consider the equation y – y 1 = m ( x – x 1 ) . In this equation, if m and x 1 are fixed and different lines are drawn for different values of y 1 , then

If the perpendicular bisector of the line segment joining P ( 1 , 4 ) and Q ( k , 3 ) has y-intercept – 4 . Then, a possible value of k , is

The equation of the pair of lines through the point ( a , b ) parallel to the coordinates axes, is

If the line 2 x − y + 3 = 0 is at distances 1 5 and from the lines 4 x − 2 y + α = 0 and 6 x − 3 y + β = 0 respectively the sum of all possible values of a and β , , is

If the straight line a x + b y + c = 0 make a triangle of constant area with coordinate axes, then

Let ABC be a triangle whose vertices, A ( 1 , − 1 ) , B ( 0 , 2 ) C x ′ , y ′ such that area of △ A B C is 5 sq . and C x ′ , y ′ lie on 3 x + y − 4 λ = 0 . Then, λ =

The incentre of the triangle with vertices ( 1 , 3 ) , ( 0 , 0 ) and ( 2 , 0 ) is

Let PS be the median of the triangle with ertices P ( 2 , 2 ) , Q ( 6 , – 1 ) and R ( 7 , 3 ) . The equation of the line passing through ( 1 , – 1 ) and parallel to P S , is

The line which is parallel to x -axis and crosses the curve y = x at an angle of 45 ∘ , is

If A and B are two fixed points, then the locus of a point which moves in such a way that the angle APB is a right angle, is

If the equation 2 x 2 + 2 h x y + 6 y 2 − 4 x + 5 y − 6 = 0 represents a pair of straight lines, then the length of intercept on the x -axis cut by the lines is equal to

Let A(1, 0), B(6, 2) and C 3 2 , 6 be the vertices of a triangle ABC. If P is a point inside the Δ A B C such that Δ A P C ,   Δ A P B   &   Δ B P C have equal areas, then the length of the line segment PQ, where Q is the point − 7 6 , − 1 3 is .

The locus of the mid-points of the perpendiculars drawn from points on the line, x=2y to the line x = y is:

Let C be the centroid of the triangle with vertices (3, -1) ,(1, 3) and (2, 4). Let P be the point of intersection of the lines x+3y-1=0 and 3x-y+1=0. Then the line passing through the points C and P also passes through the point:

A point R on the x-axis such that PR+RQ is minimum when P= (1, 1) and Q= (3, 2) is

A line is drawn from A ( – 2 , 0 ) to intersect the curve y 2 = 4 x at P and Q in the first quadrant such that 1 A P + 1 A Q < 1 4 . Then the slope of the line is always

The distance between the orthocentre and circumcentre of triangle with vertices ( 1 , 2 ) , ( 2 , 1 ) and 3 + 3 2 , 3 + 3 2 is

In △ A B C if A ( 0 , 0 ) , internal angle bisector through vertex B is x + 2 y − 5 = 0 and perpendicular bisector of the side A C is 3 x − y − 10 = 0 , then the absolute value of slope of line B C is

Two sides of a rectangle are 3x+4y+5=0 and 4x-3y+15=0. If one of its vertices is (0,0), then the area of the rectangle is

A light ray emerging from the point source placed at P(2,3) is reflected at a point Q on the y-axis. It then passes through the point R(5, l0). The coordinates of Q are

Consider the points A ( 0 , 1 ) and B ( 2 , 0 ) , and P be a point on the line 4 x + 3 y + 9 = 0 . The coordinates of P such that | P A − P B | is maximum are

The point A (2, l) is translated parallel to the line x – y = 3 by a distance of 4 units. If the new position A 1 is in the third quadrant, then the coordinates of A 1 are

The centroid of an equilateral triangle is ( 0 , 0 ) . If two vertices of the triangle lie on x + y = 2 2 , then one of them will have its coordinates

The number of equilateral triangles with y = 3 ( x − 1 ) + 2 and y = − x as two of its sides is

I f A cos α , sin α , B sin α , – cos α , C 1 , 2 a r e t h e v e r t i c e s o f a t r i a n g l e A B C t h e n a s α v a r i e s , t h e l o c u s o f t h e c e n t r o i d o f t h e t r i a n g l e i s k x 2 + y 2 – 2 x – 4 y + 1 = 0 t h e n k =

Two consecutive sides of a parallelogram are 4x + 5y = 0 and 7x + 2y = 0. If the equation to one diagonal is 11x + 7y = 9, then the equation of the other diagonal is

The image of the pair of lines represented by a x 2 + 2 h x y + b y 2 = 0 by the line mirror y = 0 is

T h e c o m b i n e d e q u a t i o n o f p a i r o f l i n e s i s 3 x 2 – 4 x y + 3 y 2 = 0 , i f t h e l i n e s w e r e r o t a t e d a b o u t t h e o r i g i n b y π 6 i n t h e a n t i c l o c k w i s e d i r e c t i o n t h e n t h e n e w p o s i t i o n i s

Two sides of a rectangle are 3x+ 4y + 5 =0 and4x-3y + 15 = 0. If one of its vertices is (0, 0), then the area of the rectangle is .

I f p i s t h e l e n g t h o f t h e p e r p e n d i c u l a r f r o m t h e o r i g i n t o t h e l i n e x a + y b = 1 t h e n

The distance between the circumcentre and orthocentre of the triangle whose vertices are (0, 0), (6,8) and (4,3) is L then the value of L 5 is

The line x = c cuts the triangle with corners (0, 0), (1, 1) and (9, 1) into two regions. For the area of the two regions to be the same, c must be equal to .

T h e e q u a t i o n o f t h e l i n e w h i c h i s e q u i d i s tan t f r o m t h e p a r a l l e l l i n e s 9 x + 6 y – 7 = 0 a n d 3 x + 2 y + 6 = 0

One diagonal of a square is3x-4y + 8 = 0 and one vertex is (-l , 1). Then the area of square is

If the distance between parallel lines 3x + 4y +7 = 0 and ax + y + b = 0 is 1 , then sum of values of b is

If the point P 3 a 2 , 1 lies between the two different lines x + y = a and x + y = 2 a , then the least positive integral value of a is

ABC is an isosceles triangle. If the coordinates of the base are. B 1 , 3 ) and, C – 2 , 7 , the coordinates of vertex A can be

Let A ≡ ( 3 , − 4 ) , B ≡ ( 1 , 2 ) . Let P ≡ ( 2 k − 1 , 2 k + 1 ) be a variable point such that PA + PB is the minimum. Then kis

If A 1 , p 2 , B ( 0 , 1 ) , and C ( 1 , 0 ) are the coordinates of three points, then the value of p for which the area of triangle A B C is the minimum is

If the point x 1 + t x 2 − x 1 , y 1 + t y 2 − y 1 divides the join of x 1 , y 1 and x 2 , y 2 internally, then

The locus of the moving point whose coordinates are given by e t + e − t , e t − e − t where t is a parameter, is

The number of integral points (x, y) (i.e., x and y are both integers) which lie in the first quadrant but not on the coordinate axes and also on the straight line 3 x + 5 y = 2007 is equal to

The incenter of the triangle with vertices 1 , 3 , (0, 0), and (2, 0) is

The orthocenter of the triangle formed by the lines x y = 0 and x + y = 1 is

The area enclosed by 2 | x | + 3 | y | ≤ 6 is

Area of the regular hexagon whose diagonal joins (2, 4) and (6, 7) is

ABCD is a square. Points E(4, 3) and F(2,5) lie on AB and CD, respectively, such that EF divides the square in two equal parts. If the coordinates of A are (7, 3), then other coordinates of the vertices can be

If A ( 5 , 2 ) , B ( 10 , 12 ) and P ( x , y ) is such that A P P B = 3 2 , then the internal bisector of ∠ A P B always passes through

If coordinates of orthocentre and centroid of a triangle are (4, -l) and (2, I ), respectively, then coordinates of a point which is equidistant from the vertices of the triangle is

The straight line ax + by + c = 0, where a b c ≠ 0 will pass through the first quadrant if

One diagonal of a square is along the line 8x – l5y=0 and one of its vertex is ( l, 2). Then the equations of the sides ofthe square passing through this vertex are

The equation of a line through the point (1, 2) whose distance from the point (3, 1) has the greatest value is

The line PQ whose equation is x -y = 2 cuts the x-axis at P, and Q is (4, 2). The line PQ is rotated about P through 45 o in the anticlockwise direction. The equation of the line PQ in the new position is

A line is drawn perpendicular to line y = 5x, meeting the coordinate axes at A and B. If the area of triangle OAB is 10 sq. units, where O is the origin, then the equation of drawn line is

The equation of the straight line which passes through the point (-4, 3) such that the portion of the line between the axes is divided internally by the point in the ratio 5:3is

The equations to a pair of opposite sides o fa parallelogram are x 2 − 5 x + 6 = 0 and y 2 − 6 y + 5 = 0 . The equations to its diagonals are

The number of possible straight lines, passing through (2, 3) and forming a triangle with co-ordinate axes, whose area is 12 sq. units is

The bisectors BD and CF of a triangle ABC have equations y = x and x = 10. If A is (3, 5), then the equation of BC is

The straight lines x + y = 0,3x + y = 0, and x + 3y – 4 = 0 form a triangle which is

The area of the parallelogram formed by the lines y = m x , y = m x + 1 , y = n x , and y = n x + 1 equals

A straight line through the origin O meets the parallel lines 4x + 2y = 9 and2x + y + 6 = 0 at points P and Q respectively. Then the point O divides the segment PQ in the ratio

The straight lines 7x – 2y + 10 = 0 and 7x + 2y – 10 = 0 form an isosceles triangle with the line y = 2. The area of this triangle is equal to

The straight lines 4ax + 3by + c = 0, where a + b + c = 0, are concurrent at the point

The lengths of the perpendicular from the points m 2 , 2 m , m m ′ , m + m ′ and m ′ 2 , 2 m ′ to the line x + y + 1 = 0 form

The image of P(a, b) on the line y = – x is Q and the image of Q on the line y = x is R. Then the midpoint of PR is

If the lines y = ( 2 + 3 ) x + 4 and y = k x + 6 are inclined at an angle 60 ∘ to each other, then the value of k will be

The distance between 4 x + 3 y = 11 and 8 x + 6 y = 15 , is

The line x a + y b = 1 meets the x -axis at A and y -axis at B and the line y = x at C such that the area of the △ A O C is twice the area of Δ B O C . Then the coordinates of C are

Area of the triangle formed by the line x + y = 3 and the angle bisectors of the pairs of straight lines x 2 − y 2 + 2 y = 1 i s

Area of quadrilateral formed by the lines 4 y − 3 x − a = 0 , 3 y − 4 x + a = 0 , 4 y − 3 x − 3 a = 0 and 3 y − 4 x + 2 a = 0 is

The set of real values of k for which the lines x + 3 y + 1 = 0 , k x + 2 y − 2 = 0 and 2 x − y + 3 = 0 form a triangle is

Number of positive integral values of b for which the origin and the point ( 1 , 1 ) lie on the same side of the straight line, a 2 x + a b y + 1 = 0 ∀ a ∈ R is

A B C is a variable triangle such that A is ( 1 , 2 ) , and B and C lie on the line y = x + λ ( λ is a variable). Then the locus of the orthocenter of △ A B C is

If the sum of the distances of a point from two perpendicular lines in a plane is 1 , then its locus is

The locus of a point that is equidistant from the lines x + y − 2 2 = 0 and x + y − 2 = 0 is

The line joining ( 5 , 0 ) to ( 10 cos ⁡ θ , 10 sin ⁡ θ ) is divided internally in the ratio 2 : 3 at P . The locus of P is

The line L 1 ≡ 4 x + 3 y − 12 = 0 intersects the x − and y -axis at A and B , respectively. A varıable ine perpendicular to L 1 intersects the x – and the y -axis at P and Q , respectively. Then the locus of the circumcenter of triangle A B Q is

Two straight lines rotate about two fixed points (-a, O) and (a,0) in anti clockwise direction. If they start from their position of coincidence such that one rotates at a rate double of the other, then locus of curve is

Consider the family of lines ( x − y − 6 ) + λ ( 2 x + y + 3 ) = 0 and ( x + 2 y − 4 ) + μ ( 3 x − 2 y − 4 ) = 0 .If the lines of these two families are at right angle to each other, then the locus of their point of intersection, is

A variable line whose slope is -2 cuts x and y-axes respectively at points A and C. If Rhombus ABCD is completed such that the vertex B lies on the line y = x , then the locus of the vertex D is

Pair of straight lines through A ( 1 , 1 ) are drawn to intersect the line 2 x + 4 y = 5 at B and C . If angle between the pair of straight line is π 3 , then the locus of incentre of Δ A B C is

The locus of the foot of the perpendicular from the origin on each member of the family ( 4 a + 3 ) x − ( a + 1 ) y − ( 2 a + 1 ) = 0 is

Distance of the origin from the line ( 1 + 3 ) y + ( 1 − 3 ) x = 10 along the line y = 3 x + k is

If the quadrilateral formed by the lines a x + b y + c = 0 , a ′ x + b ′ y + c = 0 , a x + b y + c ′ = 0 , a ′ x + b ′ y + c ′ = 0 has perpendicular diagonals, then

The number of triangles that the four lines y = x + 3, y = 2x + 3, y = 3x + 2, and y + x = 3 form is

If the straight lines x + y − 2 = 0 , 2 x − y + 1 = 0 , and a x + b y − c = 0 are concurrent, then the family of lines 2 a x + 3 b y + c = 0 ( a , b , c are nonzero ) is concurrent at

The line x + 3y – 2 = 0 bisects the angle between a pair of straight lines of which one has equation x – 7y + 5 = 0.The equation of the other line is

If a pair of perpendicular straight lines drawn through the origin forms an isosceles triangle with the line 2x + 3y = 6, then area of the triangle so formed is

Through the point P ( α , β ) , where α β > 0 , the straight line x a + y b = 1 is drawn so as to form with axes a triangle of area S . If a b > 0 , then least value of S is

Let P be the point (-3, 0) and Q be a moving point (0,3t). Let PQ be trisected at R so that R is nearer to Q. RN is drawn perpendicular to PQ meeting the x-axis at N. The locus of the mid-point of RN is

The true set of real values of “a” such that the point M ( a , sin ⁡ a ) lies inside the triangle formed by the lines x − 2 y + 2 = 0 , x + y = 0 and x − y − π = 0 is

If the vertices x i , y i , i = 1 , 2 , 3 of the triangle lie on the curve x y = 1 and x i satisfies the equation x 3 − 2 x 2 + 7 x − 1 = 0 , then the sum of coordinates of the centroid of the triangles is

If the image of P(3,5) with respect to the line y = x is Q and image of Q with respect to the line y = 0 is R(a,b), then a + b =

The lines p p 2 + 1 x − y + q = 0 and p 2 + 1 2 x + p 2 + 1 y + 2 q = 0 are perpendicular to a common line for

If the intercepts cut off on the coordinate axes by the line a x + b y + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2 x − 3 y + 6 = 0 on the co ordinate axes respectively then 25 a + b =

Two sides of a parallelogram are along the lines x + y = 3 and x − y + 3 = 0 . If its diagonals intersect at ( 2 , 4 ) then one of its vertex is

The tangent of angle between the lines x 3 + y 2 = 1 and x 3 − y 2 = 1 is

The equation of the sides of an Isosceles right angled triangle whose hypotenuse is 7 x + y − 8 = 0 and opposite vertex to the hypotenuse is 1 , 1 are

The acute angle between the lines y = 2 − 3 x − 6 and y = 2 + 3 x + 8 is

The reflection of the orthocentre of the triangle formed by the points ( 2 , 3 ) , ( 2 , 4 ) and ( 5 , 4 ) in x -axis is.

Equation of the line passing through the point ( 5 , 4 ) and making an angle 45 ∘ with negative direction of x – axis is

Equation of the line passing through the point ( 5 , 7 ) and having y -intercept 5 is

The angle made by the perpendicular drawn from the origin to the line x + 2 y − 2 5 = 0 with X – axis is

Foot of the perpendicular from ( 3 , 2 ) on the line perpendicular to y − axis and passing though the origin is

Suppose that A ( 3 , 5 ) and B ( 7 , 1 ) are any two given points. The point where the line with slope m = 1 and bisecting the line segment A B ¯ meets the line A B ¯ is

The foot of the perpendicular drawn from the point 5 , 4 on the line whose equation is X = 0

For the line x + y − 2 = 0 1 , 2 , 1 , − 2 , the points

The equation of base of an equilateral triangle is 4 x + 3 y − 25 = 0 and one vertex is origin then the length of the side is

The perpendicular distance from 1 , 3 to the line passing through the point P 2 , 1 where the point P 2 , 1 bisects the portion of the line intercepted between the axes.

If A and B are two points on the line 12 x + 5 y + 65 = 0 , such that O A = O B = 13 then the area of the triangle is

One vertex of an equilateral triangle is 2 , − 1 whose area is 1 2 3 .The perpendicular distance from the vertex 2 , − 1 to opposite side is

If 9 a 2 + 25 b 2 − c 2 + 30 a b = 0 , then the set of lines a x + b y + c = 0 passes through the fixed point is

Equation of the line passing through the point of intersection of x + y − 4 = 0 and 2 x + 3 y − 10 = 0 , parallel to the line 5 x − 7 y + 5 = 0

The triangle formed by the lines 2 x + 3 y − 5 = 0 , x − y + 4 = 0 and 6 x − 4 y + 1 = 0 is

The combined equation of angular bisector of lines represented by the equation x y = 0 is

The triangle formed by the lines given by 3 x + y = 0 , 3 x − y = 0 , and y − 2 = 0

The combined equation of three lines is x 2 − y 2 2 x + 5 y + 7 = 0 ,the number of circles which touches all three lines is

Angle between the angular bisector of two lines is

If two lines 4 x + 3 y + 5 = 0 and l x + m y + 4 = 0 are the angular bisectors of two lines then the relation between l , m is

A straight line L = 0 passes through the point 5 , − 4 is inclined at an angle of 60 ° to the line 3 x + y − 3 = 0 ,then the equation of the line L = 0 is

Slope of a line L is 1. The points on the line L at a distance of 2 units from the point 3 2 , 3 2 which lies on the line L are

The distance between 1 , 1 and the point of intersection of two lines 2 x + 3 y + c = 0 and 3 x + 2 y + c = 0 is less than 2 2 then

Suppose that A 1 , 2 and B 3 , 4 be two points and P be a point on y = x such that P A + P B is minimum then the point P

Suppose that A 5 , 6 and B 3 , 4 be two points and P be a point on x – axis such that P A + P B is minimum then the point P is

Acute angle between the lines whose slopes are m 1 = 2 , m 2 = 1 2 is θ then the value of tan θ is

Suppose that A 5 , 1 and B 7 , 5 be two points and P be a point on y = x such that P A − P B is minimum then the point is P

Acute angle, in radians between the lines whose slope is 2 and the line which is perpendicular to the line 2 x − y + 5 = 0 is

If 3 x − 4 y + 5 = 0 and a x + 2 y + 4 = 0 are parallel to each other then a =

The inclination of the line which is angular bisector of coordinate axes in the second quadrant is (in radians)

The image of centroid of the triangle formed by the points 0 , 0 , 1 , 2 , 2 , 1 in y – axis is h , k then h + k is

Reflection of the midpoint of the line segment joining the points 1 , 2 and 3 , 4 with respect to the line y = x is a , b . The value of a + b is

The normal form of the line x + y + 1 = 0 is x cos ⁡ α + y sin ⁡ α = p then α =

The foot of the perpendicular from the point 1 , 2 on the line passing through the points 5 , 4 and 7 , 4 is a , b then a + b =

The point where the perpendicular from the point − 5 , 7 meets the line 3 x + 4 y + 12 = 0 is h , k then the value of h 2 k + 2 is

OPQR is a square and M, N are the middle points of the sides PQ and QR , respectively. Then the ration of the area of the square to that of triangle OMN is

A particle is moving in a straight line and at some moment it occupied the positions (5,2) and (-1,2). Then the position of the particle when it is on x-axis is

Let A ≡ ( 3 , − 4 ) , B ≡ ( 1 , 2 ) .   Let   P ≡ ( 2 k − 1 , 2 k + 1 ) be a variable point such that PA+PB is the minimum. Then k is

The polar coordinates equivalent to ( − 3 , 3 ) are

On vertex of an equilateral triangle is (2,2) and its centroid is ( − 2 3 , 2 3 ) . The length of its side is

If (2,-3),(6,5) and (-2,1) are three consecutive vertices of a rhombus, then its area is

If points A(3,5) and B are equidistant from H ( 2 , 5 ) and B has rational coordinates, then AB=

Let n be the number of points having rational coordinates at a fixed distance from the point ( 0 , 3 ) . Then

In ΔABC , the sides BC=5, CA=4 and AB=3. If A ≡ ( 0 , 0 ) and the internal bisector of angle A meets BC in D ( 12 7 , 12 7 ) , then incentre of ΔABC is

If two vertices of a triangle are (-2,3) and (5,-1), the orthocentre lies at the origin, and the centroid on the line x+y=7, then the third vertex lies at

The vertices of a triangle are ( pq , 1 / ( pq ) ) , ( qr , 1 / ( qr ) )   and    ( rq , 1 / ( rq ) ) , where p,q and r are the roots of the equation y 3 − 3 y 2 + 6 y + 1 = 0 . The coordinates of its centroid are

Two vertices of a triangle are (4,-3) and (-2,5). If the orthocenter of the triangle is at (1,2), then the third vertex is

A triangle ABC with vertices A ( − 1 , 0 ) , B ( − 2 , 3 / 4 ) , C ( − 3 , − 7 / 6 ) has its orthocenter at H . Then, the orthocenter of triangle BCH will be

The point P (2,1) is translated parallel to the line L: x-y = 4 by 2 3 units. If the new point Q lies in the third quadrant, then the equation of the line passing through Q and perpendicular to L is

The number of integer value of m, for which the x-coordinate of the point of intersection of the lines 3x+4y = 9 and y = mx + 1 is also an integer is

Slope of a line L is 1. The points on the line L at a distance of 2 units from the point 3 2 , 3 2 which lies on the line L are

A straight line L = 0 passes through the point 5 , − 4 is inclined at an angle of 60 o to the line 3 x + y − 3 = 0 ,then the equation of the line L = 0 is

The distance between 1 , 1 and the point of intersection of two lines 2 x + 3 y + c = 0 and 3 x + 2 y + c = 0 is less than 2 2 then

Suppose that A 5 , 6 and B 3 , 4 be two points and P be a point on x- axis such that PA + PB is minimum then the point P is

Suppose that A 1 , 2 and B 3 , 4 be two points and P be a point on y=x such that PA + PB is minimum then the point P

Acute angle between the lines whose slopes are m 1 = 2 , m 2 = 1 2 is θ then the value of tanθ is

Acute angle, in radians between the lines whose slope is 2 and the line which is perpendicular to the line 2 x − y + 5 = 0 is

If 3 x − 4 y + 5 = 0 and ax + 2 y + 4 = 0 are parallel to each other then a=

The tangent of angle between the lines whose intercepts on the axes are 4 , − 5 and 5 , − 4 respectively

The inclination of the line passing through the point of intersection of lines given by y = x and the point 2 , 0 is

Equation of one of the line through the point 1 , 5 and making an angle of 45 ° with the line x − 3 y + 5 = 0 is ax + by + c = 0 . If a > 0 , b > 0 then the value of a + b + c is

The inclination of the line which is angular bisector of coordinate axes in the second quadrant is (in radians)

The image of centroid of the triangle formed by the points 0 , 0 , 1 , 2 , 2 , 1 in y- axis is h , k then h + k is

Reflection of the midpoint of the line segment joining the points 1 , 2 and 3 , 4 with respect to the line y = x is a , b . The value of a + b is

Reflection of the point 1 , 1 with respect to the line 4 x + 3 y − 5 = 0 is α , β then α + β

If the line 4 x − 5 y + 41 = 0 is the perpendicular bisector of the line segment joining the points A 1 , 1 and B .The image of B in x-axis is a , b then the numerical value of a + b is

The perpendicular distance from origin to the line 4 x + 3 y − 5 = 0 is

If the equation k 2 x 2 + 2 xy + 9 y 2 = 0 represents a pair of distinct lines, then k lies in the internal

The combined equation of straight lines passing through point − 1 , 2 and each line makes intercepts on the coordinate axes whose product is

The equation sin 2 θx 2 + 2 kxy + cos 2 θy 2 = 0 represents coincident lines then k is

If k 1 , k 2 are the sum and product of the slopes of the pair of lines 2 x 2 + 4 xy + y 2 = 0 then k 1 + k 2 is

The equation 3 x 2 + 4 xy + ky 2 = 0 represents distinct lines then k is

The normal form of the line x + y + 1 = 0 is x cos α + y sin α = p then α =

The foot of the perpendicular from the point 1 , 2 on the line passing through the points 5 , 4 and 7 , 4 is a , b then a + b

If one of the lines 4 x 2 − cxy + y 2 = 0 is x + y = 0 then c =

The equation lx 2 + 2 mxy + ny 2 = 0 represents a pair lines then the relation between l , m is

If θ is acute angle between two lines x 2 + 6 xy + y 2 = 0 then the value of 2 sinθ + cosθ is

If a,b are intercepts of the line which is perpendicular to the line passing through the points 3 , 5 , 4 , 3 and passing through the point 5 , 6 then a b =

The x- intercept of the line joining the points 3 , 1 and 1 , 3 is

The point where the perpendicular from the point − 5 , 7 meets the line 3 x + 4 y + 12 = 0 is h , k then the value of h 2 k + 2 is

The equations of lines whose intercepts are the roots of the equation x 2 − 3 x + 2 = 0 are

If the pair of lines represented by the combined equation 2 x 2 + 4 xy + y 2 = 0 makes angles θ 1 , θ 2 with x − axis then cos θ 1 − θ 2 =

The intercepts made by the line x − y = k with axes are

The length of portion of line x + y − 2 = 0 intercepted between the axes is

The sum of the intercepts of the line whose portion intercepted between the axes bisected by

The equations of lines are given by sinθ x + a = 1 + cosθ y , sinθ x − a = − 1 − cosθ y . The locus of point of intersection of lines is x m + y n = a l then l + m + n is

The Locus of the midpoint of the portion of the line 3 xsecθ + 4 ytanθ = 1 intercepted between the axes is 1 ax 2 − 1 by 2 = 1 then a + b =

Consider the family of lines x − y − 6 + λ 2 x + y + 3 = 0 and x + 2 y − 4 + μ 3 x − 2 y − 4 = 0 . If the lines of these two families intersect at right angles to each other, then the locus of their point of intersection is a circle with radius

The sum of the slopes of lines which are parallel, perpendicular to the line passing through the points 1 , 3 and 3 , 5 is

Suppose that a,b,c are in arithmetic progression then the line ax + by + c = 0 passes through the fixed point l , m then l + m =

If the lines 2 x + y − k = 0 , x − y + 1 = 0 and − 2 x − 3 y + 8 = 0 are concurrent then k=

The lines 2 x + 5 y = 10 and 3 x + 4 y = 12 cuts the x-axis at A and B respectively. A line L drawn through the point 3 , 4 meets y-axis at C such that the abscissae of A ,B, and ordinate of C are in geometric progression. If the equation of the line L is ax + by + c = 0 then a + b + c is

The area of the quadrilateral formed by the lines 2 x + 3 y + 5 = 0 , 3 x − 2 y + 4 = 0 , 4 x + 6 y − 4 = 0 and 6 x − 4 y + 3 = 0 is ( in square units )

If a and b are slopes of the angular bisectors of the coordinate axes then a 2 + b 2 is

The equation of angular bisector of the lines 3 x + 4 y + 5 = 0 and 4 x − 3 y + 7 = 0 is lx + my + n = 0 , which makes acute angle with x-axis then the value of l + m + n is

The angle between the pair of lines x 2 + 4 xy + 2 y 2 = 0 is

Number of triangles formed by the lines a x + y − 2 + b 3 x + 4 y + 7 = 0 and 5 x − 4 y − 1 = 0

The angle between the pair of lines cos 2 θ x 2 + 2 xysinθ + sin 2 θ − 1 = 0 is

Two lines are intersecting at the point 1 , 1 . If 2 x + 3 y − 5 = 0 is the equation of one angular bisector of the two lines and the equation of the other bisector is lx + my + n = 0 then l + m + n is

The angle between the pair of lines cos 2 θ x 2 + 2 xy sinθ + sin 2 θ − 1 = 0 is

Let the line L=0 makes intercepts 4,5 on axes, then the area of the triangle formed by the line L=0 with coordinate axes is ( in square units )

The tangent of the acute angle between the pair of lines 2 x 2 − 3 xy + y 2 = 0 is

The combined equation of pair of lines passing through the point 1 , 2 and parallel to the pair of lines x 2 + 2 xy + y 2 = 0

If the lines represented by the equations 2 x 2 + mxy + y 2 + 3 x + ly − 5 = 0 are parallel to each other, which of the following is correct.

The locus of midpoints of the perpendicular drawn from the points on the line x = 3 y to the line x=y is

If the pair of lines x 2 + 2 xy + ky 2 = 0 are perpendicular then k =

The combined equation 5 x 2 + 4 xy − 5 y 2 = 0 represents the pair of lines and those lines are

If the pair of lines x 2 cosθ + 3 xytanθ − y 2 sinθ = 0 are perpendicular to each other then θ

If 2 x 2 + 3 xy + ky 2 − 5 x + 5 y + 7 = 0 represents a pair of perpendicular lines, k =

The equation to the pair of lines perpendicular to the lines represented by the equation x 2 + 4 xy + 2 y 2 = 0 and passing through origin is 2 x 2 + kxy + y 2 = 0 then k =

The equation to the pair of lines perpendicular to x 2 + xy + y 2 = 0 and passing through (2,1),is x 2 − xy + y 2 + lx + my + n = 0 then l + m + n =

If A p , p 2 falls inside the angle made by the lines x − 2 y = 0 and 3 x − y = 0 , x > 0 then p belong to

The equation to the pair of lines perpendicular to 2 x 2 + 3 xy + y 2 = 0 and passing through 2 , 1 is x 2 − 3 xy + 2 y 2 + lx + my + n = 0 then l + m + n =

The equation to the pair of lines passing through (1,2)and perpendicular to x 2 + 2 xy + y 2 = 0 is

The product of the perpendicular from 1 , 2 to the pair of lines x 2 + 2 xy + y 2 = 0 is

The product of the perpendicular from (-1,2) to the pair of lines 2 x 2 + kxy + y 2 = 0 is 2 then the value of k is

The product of the perpendicular from 1 , 1 to the pair of lines x 2 + 2 xy + y 2 + 2 x + 2 y + 1 = 0 is

The product of the perpendicular from (1,1) to the pair of lines x 2 + 2 xy + y 2 + kx + 2 y + 5 = 0 is 2 then k = k < 0

If O , A , B are vertices of a triangle whose sides are given by 5 x + 12 y 2 − 3 12 x − 5 y 2 = 0 and 5 x + 12 y − 78 = 0 then AB =

The product of the perpendicular from origin to the pair of lines 5 x 2 + 12 xy + 6 y 2 + x + y − 7 = 0 is

The product of the perpendicular from (1,1) to the pair of lines ax 2 − 2 hxy + by 2 = 0 is 1 then 1 a + 1 b

The product of perpendicular from (1,1) to the pair of lines 2 x 2 + 3 xy + y 2 = 0 is equal to the product of perpendiculars from (0,0) to the pair of lines x 2 + 2 xy + 3 y 2 + x + 2 y + k = 0 then k =

The area of the triangle formed by the lines x 2 + 3 xy + y 2 = 0 , x + y + 1 = 0 is

The area of the triangle formed by the lines 3 x 2 − 4 xy + y 2 = 0 , 2 x − y + 6 = 0 is

The area of the triangle formed by the lines y = x + k , ax 2 + 2 hxy + by 2 = 0 is

The base of an equilateral triangle is along the line given by 3 x + 4 y = 9 . If a vertex of the triangle is 1 , 2 then the length of its side is

The circumcentre of a triangle lies at the origin and the centroid is the midpoint of the line segment joining the points a 2 + 1 , a 2 + 1 and ( 2 a , − 2 a ) , then the orthocentre lies on the line.

For all real values of a and b 2 a + b x + a + 3 b y + b − 3 a = 0 and mx + 2 y + 6 = 0 are concurrent, then m is equal to

Let A , B be the feet of the perpendiculars drawn from C 1 , 1 to the pair of lines 7 x 2 − 20 xy + 12 y 2 = 0 . If P is the midpoint of OC then PA + PB =

The circumcentre of the triangle formed by the lines x 2 − y 2 = 0 and y − 5 = 0 is

If the shortest distance between the lines x + 2 λ = 2 y = − 12 z , x = y + 4 λ = 6 z − 12 λ is 4 2 units, then a value of λ is

The area of the triangle formed by pair of lines ax + by 2 – 3 bx – ay 2 = 0 and the line ax + by + c = 0 is

The area of the equilateral triangle formed by the pair of lines passing through the origin with the line 3 x + 4 y – 5 = 0

A light beam from the point A(3,10) reflects from the straight line 2 x + y − 6 = 0 and then passes through the point B(7,2) . Find the equations of the incident and reflected beams.

If a + b = 2 h , then the area of the triangle formed by the lines ax 2 + 2 hxy + by 2 = 0 and the line x – y + 2 = 0 in square units is

Equation, 2 x 2 – 5 xy + 2 y 2 = 0 represents two sides of a triangle whose centroid is (2,3) then area of triangle is

If the lines x + y + 1 = 0 , 4 x + 3 y + 4 = 0 and x + αy + β = 0 where α 2 + β 2 = 2 are concurrent then

If the equations of the sides of a triangle are x + y = 2, y = x and 3 y + x = 0 , then which of the following is an exterior point of the triangle?

If the vertices P, Q, R of a ∆ PQR are rational points, which of the following points of the ∆ PQR is (are) always rational point(s)?

If α + β + γ = 0 , the line 3 α x + β y + 2 γ = 0 passes through the fixed point

The line (p + 2q) x + (p – 3q)y = p – q for different values of p and q passes through the fixed point

The point (3, 2) is reflected in the y-axis and then moved a distance 5 units towards the negative side of y-axis. The coordinates of the point thus obtained are

The number of integer values of m, for which the x-co-ordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer, is

If a, b, c are three terms of an A.P., then the line ax + by + c = 0

The diagonals of the parallelogram whose sides are lx + my + n = 0, lx + my + n′ = 0, mx + ly + n = 0, mx + ly + n′ = 0 include an angle

A ray of light is sent along the line which passes through the point (2, 3). The ray is reflected from the point P on x-axis. If the reflected ray passes through the point (6, 4), then the coordinates of P are

If P and Q are two points on the line 4x + 3y + 30 = 0 such that OP = OQ = 10, where O is the origin, then the area of the ∆ OPQ is

On the portion of the straight line x + y = 2 which is intercepted between the axes, a square is constructed away from the origin, with this portion as one of its side. If p denotes the perpendicular distance of a side of this square from the origin, then the maximum value of p is

If 1 a , 1 b , 1 c are in A.P., then the straight line x a + y b + 1 c = 0 always passes through a fixed point, that point is

Let P(2, 0) and Q(0, 2) be two points and O be the origin. If A(x, y) is a point such that xy > 0 and x + y < 2, then

Number of equilateral triangles with y = 3 x – 1 + 2 and y = 3 x as two of its sides, is

The vertices of a ∆ OBC are O (0, 0), B(–3, –1) and C(–1, –3). The equation of a line parallel to BC and intersecting sides OB and OC whose distance from the origin is 1 2 , is

A line joining two points A(2, 0) and B(3, 1) is rotated about A in anticlockwise direction through an angle 15°. If B goes to C in the new position, then the coordinates of C are

If the equal sides AB and AC (each equal to a) of a right angled isosceles triangle ABC be produced to P and Q so that B P × C Q = A B 2 , then the line PQ always passes through the fixed point

The image of the point (–8, 12) with respect to the line mirror 4x + 7y + 13 = 0 is

Given the system of straight line a(2x + y – 3) + b(3x + 2y – 5) = 0, the line of the system farthest form the point (4, –3) has the equation

In the above problem, coordinates of the point P such that |PA – PB| is minimum are

A square is constructed on the portion of the line x + y = 5 which is intercepted between the axes, on the side of the line away from origin. The equations to the diagonals of the square are

The image of the point P(3, 5) with respect to the line y = x is the point Q and the image of Q with respect to the line y = 0 is the point R(a, b), then (a, b)

A line L has intercepts a and b on the coordinate axes. When the axes are rotated through an angle, keeping the origin fixed, the same line L has intercepts p and q. Then,

A ray of light travelling along the line x + 3y = 5 is incident on the x-axis and after refraction it enters the other side of the x-axis by turning  π 6 away from the x-axis. The equation of the line along which the refracted ray travels is

Let P be the image of the point (–3, 2) with respect to x-axis. Keeping the origin as same, the coordinate axes are rotated through an angle 60° in the clockwise sense. The coordinates of point P with respect to the new axes are

Two vertices of a triangle are (2, –1) and (3, 2) and third vertex lies on the line x + y = 5. If the area of the triangle is 4 units then third vertex is

Let P(2, – 4) and Q(3, 1) be two given points. Let R (x, y) be a point such that (x – 2) (x – 3) + (y – 1) (y + 4) = 0. If area of ∆ PQR is 13 2 , then the number of possible positions of R are

If two vertices of an equilateral triangle have integral coordinates then the third vertex will have

P(3, 1), Q(6, 5) and R(x, y) are three points such that the angle RPQ is a right angle and the area of ∆ RPQ = 7, then the number of such points R is

Through the point P ( α , β ) , where α β > 0 the straight line x a + y b = 1 is drawn so as to form with coordinate axes a triangle of area S. If α β > 0 , then the least value of S is

If the centroid and a vertex of an equilateral triangle are (2, 3) and (4, 3) respectively, then the other two vertices of the triangle are

If a triangle has its orthocentre at (1, 1) and circumcentre at 3 2 , 3 4  then the coordinates of the centroid of the triangle are

If a, b, c are in A.P., then the straight line ax + by + c = 0 will always pass through a fixed point whose coordinates are

Let ax + by + c = 0 be a variable straight line, where a, b and c are first, third and seventh terms of an increasing A.P. Then, the variable straight line always passes through a fixed point which lies on

If the points P i = a i 3 a i – 1 , a i 2 – 3 a i – 1 , i = 1 , 2 , 3 , , are collinear and a 1, a 2 and a 3 are distinct real numbers, then

If the point (2 cos θ , 2 sin θ ) does not fall in that angle between the lines y = |x – 2| in which the origin lies then θ belongs to

A straight line L with negative slope passes through the point (8, 2) and intersects the positive coordinate axes at points P and Q. As L varies the absolute minimum value of OP + OQ is (O is origin).

If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is

A ladder of length ‘a’ rests against the floor and a wall of a room. If the ladder begins to slide on the floor, then the locus of its middle point is

The distance between the circumcenter and the orthocenter of the triangle whose vertices are (0, 0), (6, 8), and (4,3) is Z. Then the value of L 5 is .

A triangle ABC has vertices A(5, 1), B(-1, -7), and C(1,4),respectively. L be the line mirror passing through C and parallel to AB. Alight ray emanating from point A goes along the direction of the internal bisector of angle A, which meets the mirror and BC at E and D, respectively. Then the sum of the areas of △ ACE and ΔABC is .

The maximum area of the convex polygon formed by joining the points A(0, 0), B(2t 2 ,0), C(18,2), D 8 t 2 , 4 and E(0,2), where t ∈ R − { 0 } and interior angle at vertex B is greater than or equal to 90° is .

A man starts from the point P(-3 ,4) and reaches the point Q (0, 1) touching the x-axis at R( α ,0) such that PR + RQ is minimum. Then α = .

If three points ( h , 0 ) , ( a , b ) and ( o , k ) lie on a line then

Equation of the line which makes an intercept of length 2 on positive x – axis and an intercept of length 3 on the negative y – axis is

If the point ( a , b ) divides a line between the axes in the ratio 2 : 3 , the equation of the line is

If the sum of the slopes of the lines given by x 2 − 2 c x y − 7 y 2 = 0 is four times their product, then the value of c is

Let A ( 2 , – 3 ) and B ( – 2 , 1 ) be vertices of a triangle A B C . If the centroid of this triangle moves on the line 2x + 3y = 1, then the locus of the vertex C is the line

The equations of the sides of a parallelogram are x = 2 , x = 3 and y = 1 , y = 5 .Equations to the pair of diagonals are

Locus of mid point of the portion between the axes o x cos ⁡ α + y sin ⁡ α = p where p is constant is

The equation x 3 − 6 x 2 y + 11 x y 2 − 6 y 3 = 0 represent three straight lines passing through the origin, the slops of which form

The lines joining the origin to the points of intersection of 3 x 2 + λ x y − 4 x + 1 = 0 and 12 x + y − 1 = 0 are at right angles for

The line joining the points ( 2 , x ) and ( 3 , 1 ) is perpendicular to the line joining the points ( x , 4 ) and ( 7 , 5 ) . The value of x is

The line L is given by x 5 + y b = 1 passes through the point ( 13 , 32 ) . The line K is parallel to L and has the equation x c + y 3 = 1 Then the distance between L and K is

Locus of the centroid of the triangle whose vertices are ( a cos ⁡ t , a sin ⁡ t ) , ( b sin ⁡ t , − b cos ⁡ t ) and (1, 0); where t is a parameter is

If x 1 , x 2 , x 3 and y 1 , y 2 , y 3 are both in G.P. with the same common ratio, then the points x 1 , y 1 x 2 , y 2 and x 3 , y 3

P is a point on x – axis, Q is a point on y-axis. Both are equidistant from the points ( 7 , 6 ) and ( 3 , 4 ) . Distance between P and Q is

Consider three points P ( cos ⁡ α , sin ⁡ β ) , Q ( sin ⁡ α , cos ⁡ β ) and R ( 0 , 0 ) . Where 0 < α , β < π / 4 Then

The equations of the pairs of opposite sides of a rectangle are x 2 − 7 x + 6 = 0 and y 2 − 14 y + 40 = 0 the equation of the diagonal nearer the origin is

The slope of a line is 3 times the slope of the other line and the tangent of the angle between them is 4 / 13 , sum of their slopes is equal to

A line perpendicular to the line segment joining the points ( 7 , 3 ) and ( 3 , 7 ) divides it in the ratio 1 : 3 , the equation of the line is

The line L has intercepts a and b on the coordinate axes. The coordinate axes are rotated through a fixed angle, keeping the origin fixed. If p and q are the intercepts of the line L on the new axes, then 1 a 2 − 1 p 2 + 1 b 2 − 1 q 2 is equal to

Lines a x + b y + c = 0 and 3 a + 2 b + 4 c = 0 a , b , c ∈ R are concurrent at the point.

If the distance between the parallel lines 3 x + 4 y + 7 = 0 and a x + y + b = 0 is 1 , the integral value of b is

If the sum of the slopes of the lines given by 3 x 2 − 2 c x y − 5 y 2 = 0 is twice their product, then the value of c is

If α , β , γ are the real roots of the equation x 3 − 3 p x 2 + 3 q x − 1 = 0 , then the centroid of the triangle with vertices α , 1 α , β , 1 β and γ , 1 γ is at the point

If P and Q are the lengths of the perpendiculars from the origin to the lines x cos θ + y sin θ = k cos 2 θ and x s e c θ – y cos e c θ = k respectively then

Image of the point (–1, 3) with respect to the line y = 2 x is

If the lines 3 x – y + 1 = 0 and x – 2 y + 3 = 0 are equally inclined to the line y = m x , then the value of m is given by

If algebraic sum of distances of a variable line from points (2, 0), (0, 2) and (– 2, – 2) is zero, then the line passes through the fixed point

The distance of the point ( 2 , 3 ) from the line 4 x – 3 y + 26 = 0 is same as its distance from the line 3 x – 4 y + p = 0 . The value of p can be

An equation of a straight line passing through the inter-section of the straight lines 3 x – 4 y + 1 = 0 and 5 x + y – 1 = 0 and making non-zero, equal intercepts on the axes is

The points (0, 8/3), (1, 3) and (82, 30) are the vertices of

The vertices of the triangle A B C are A ( 1 , 2 ) , B ( 0 , 0 ) and C ( 2 , 3 ) , then the greatest angle of the triangle is

A B C D is a square in which A lies on the positive y -axis and B lies on the positive x -axis. If D is the point (12, 17), the coordinates of C are

A ray of light passing through the point ( 3 , 7 ) reflects on the x – axis at a point A and the reflected ray passes through the point ( 2 , 5 ) , the coordinates of A are

If (0, 1), (1, 1) and (1, 0) are the mid points of the sides of a triangle, the coordinates of its incentre are

If the segment of the line between the lines x – y + 2 = 0 and x + y + 4 = 0 is bisected at the origin, equation of the line is

A line passing through the point P(1, 2) meets the line x + y = 7 at the distance of 3 units from P. Then the slope of this line satisfies the equation

A line has intercepts a and b on the coordinate axes. When the axes are rotated through an angle α , in the anti clockwise direction keeping the origin fixed, the line makes equal intercepts on the coordinate axes, then tan α =

Equations to the sides of a triangle are x – 3 y = 0 , 4 x + 3 y = 5 and 3 x + y = 0 The line 3 x – 4 y = 0 passes through the

Locus of the mid-points of the intercepts between the coordinate axes by the lines passing through ( a , 0 ) does not intersect

If h denotes the arithmetic mean and k denotes the geometric mean of the intercepts made on the coordinate axes by the lines passing through the point (1, 1), then the point ( h , k ) lies on

If a , b , c are in A.P., a, x, b are in G.P. and b, y, c are in G.P., the point (x, y) lies on

The points ( a , b + c ) , ( b , c + a ) and ( c , a + b ) are

Distance between P ( x 1 , y 1 ) and Q ( x 2 , y 2 ) when PQ is parallel to y-axis is

If A x 1 , y 1 , B x 2 , y 2 , C x 3 , y 3 are the vertices of a triangle, then the equation x      y      1 x 1      y 1      1 x 2      y 2      1 + x      y      1 x 1      y 1      1 x 3      y 3      1 = 0 represents

A straight line L through the point ( 3 , − 2 ) is inclined at an angle 60 ° to the line 3 x + y = 1 . If L If L also intersects the x-axis, then the equation of L is

Given four lines whose equations are x + 2 y − 3 = 0 , 2 x + 3 y − 4 = 0 , 3 x + 4 y − 7 = 0 and 4 x + 5 y − 6 = 0 , then the lines are

If the lines x + 2 a y + a = 0 , x + 3 b y + b = 0 and x + 4 c y + c = 0 are concurrent, then a , b , c are in

Two adjacent sides of a parallelogram are 4 x + 5 y = 0 and 7 x + 2 y = 0 . If an equation to one of the diagonals is 11 x + 7 y – 9 = 0 , then an equation of the other diagonal is

The lines parallel to the axes and passing through the point ( 4 , – 5 ) are

Points (8, 2), (– 2, – 2) and (3, 0) are the vertices of

If p is the length of the perpendicular from the origin on the line x a + y b = 1 and a 2 , p 2 , b 2 are in A.P., then a 4 – 2 p 2 a 2 + 2 p 4 =

A straight line through the origin O meet the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at points P and Q respectively. The point O divides the segment P Q in the ratio

Given four lines with equations x + 2 y − 3 = 0 , 3 x + 4 y − 7 = 0 , 2 x + 3 y − 4 = 0 , 4 x + 5 y − 6 = 0

If a , x 1 , x 2 are in G.P. with common ratio r , and b , y 1 , y 2 are in G.P. with common ratio s where s – r = 2 , then the area of the triangle with vertices ( a , b ) , ( x 1 , y 1 ) and ( x 2 , y 2 ) is

The point ( 4 , 1 ) undergoes the following three transformations successively. (i) reflection about the line y = x (ii) transformation through a distance 2 units along the positive direction of x- axis (iii) rotation through an angle p/4 about the origin in the counter clockwise direction. Then the final position of the point is given by the coordinates

The equation of the line whose perpendicular distance from the origin is 3 units and the angle which the normal makes with the positive direction of x-axis is 30° is

If the angle between the lines 3 y – x + 4 = 0 and x + y – 6 = 0 is q, then tan q is equal to.

A 1 , A 2 … A n are points on the line y = x lying in the positive quadrant such that O A n = n O A n − 1 , O being the origin. If O A 1 = 1 and the coordinates of A n are ( 2520 2 , 2520 2 ) , then n =

If the circumcentre of a triangle lies at the origin and the centroid is the middle point of the line joining the points ( a 2 + 1 , a 2 + 1 ) and ( 2 a , – 2 a ) ; then the orthocentre lies on the line

The number of integer value of m , for which the x-coordinate of the point of intersection of the lines 3 x + 4 y = 9 and y = m x + 1 is also an integer is

If x 1 , x 2 , x 3 are the abscissa of the points A 1 , A 2 , A 3 respectively where the lines y = m 1 x , y = m 2 x , y = m 3 x meet the line 2 x − y + 3 = 0 such that m 1 , m 2 , m 3 are in A.P., then x 1 , x 2 , x 3 , are in

The diagonals of a parallelogram PQRS are along the lines 8 + 3 y = 4 and 6 x − 2 y = 7 .Then P Q R S must be a

If p 1 , p 2 denote the lengths of the perpendiculars from the origin on the lines x s e c a + y cos e c a = 2 a and x cos a + y sin a = a cos 2 a respectively,

The incentre of the triangle with vertices ( 1 , 3 ) ,(0, 0) and (2, 0) is

A line is drawn through the point (1, 2) to meet the coordinate axes at P and Q such that it forms a triangle O P Q , where O in the origin. If the area of the triangle O P Q , is least, then the slope of the line P Q is

If the line 2 x + y = k passes through the point which divides the line segment joining the points (1,1) and (2,4) in the ratio 3:2, then k equals

The locus of the point of intersection of the lines x sin θ + ( 1 – cos θ ) y = a sin θ and x sin θ – ( 1 + cos θ ) y + a sin θ = 0 is

The angle which the normal to the line x – 3 y + 8 = 0 passing through the origin, makes with the positive x-axis is

If x + 2 y + 3 = 0 , x + 2 y − 7 = 0 and 2 x − y − 4 = 0 form three sides of a square, the equation of the fourth side nearer the point (1, – 1) is

Two points ( a , 3 ) and ( 5 , b ) are the opposite vertices of a rectangle. If the other two vertices lie on the line y = 2 x + c which passes through the point ( a , b ) then the value of c is

Equation of a line passing through the intersection of the lines 7x – y + 2 = 0 and x – 3y + 6 = 0 parallel to x-axis is

If the lines x = k ; k = 1 , 2 … , n meet the line y = 3 x + 4 at the points A k x k , y k , k = 1 , 2 , … , n then the ordinate of the centre of Mean position of the points A k , k = 1 , 2 , … , n is

A ray of light travels along the line 2 x − 3 y + 5 = 0 and strikes a plane mirror lying along the line x + y = 2 The equation of the straight line containing the reflected ray is

Let P Q R be a right angled isosceles triangle right angled at P (2, 1). If the equation of the line Q R is 2 x + y = 3 , then the equations representing the pair of lines P Q and P R is

Area of the triangle formed by the lines y – x = 0, x + y = 0 and y = k in square units is

The value of p for which the lines 2 x + y – 3 = 0 , 3 x – y – 2 = 0 and x – p y + 5 = 0 may intersect at a point is

If the lines joining the origin to the intersection of the line y = m x + 2 and the curve x 2 + y 2 = 1 ae at right angles, then

The distance between the parallel lines given by ( x + 7 y ) 2 + 4 2 ( x + 7 y ) − 42 = 0 is

Equations of the straight lines passing through the point ( 4 , 3 ) and making intercepts on the coordinate axes whose sum is – 1 are

The sum of the intercepts cut off by the axes on the lines x + y = a , x + y = a r , x + y = a r 2 , … where a ≠ 0 and r = 1 / 2 is

The points (k – 1, k + 2), (k, k + 1), (k + 1, k) are collinear for

If the straight lines x + 2 x − 9 = 0 , 3 x + 5 y − 5 = 0 and a x + b y + 1 = 0 are concurrent, then the straight line 35 x − 22 y − 1 = 0 passes through

If one of the lines given by the equation 2 x 2 + a x y + 3 y 2 = 0 coincide with one of those given by 2 x 2 + b x y − 3 y 2 = 0 and the other lines represented by them be perpendicular, then

The area of the triangle formed by the points (k, 2 – 2k), (– k + 1, 2k) and (– 4 – k, 6 – 2 k) is 70 units. For

Two of the lines represented by x 3 − 6 x 2 y + 3 x y 2 + d y 3 = 0 are perpendicular for

The distance of the line 2x + 3y – 5 = 0 from the point (3, 5) along the line 5x – 3y = 0 in units is

The quadrilateral ABCD formed by the points A (0, 0); B (3, 4), C (7, 7) and D (4, 3) is a

The triangle with vertices A ( 2 , 7 ) , B ( 4 , y ) and C ( – 2 , 6 ) is right angled at A if

The line parallel to x – axis passing through the intersection of the lines a x + 2 b y + 3 b = 0 and b x – 2 a y – 3 a = 0 where ( a , b ) ≠ ( 0 , 0 ) is

Line passing through the point P ( 2 , 3 ) meets the lines represented by x 2 − 2 x y − y 2 = 0 at the points A and B such that P A . P B = 17 , the equation of the line is

The join of the points (– 3, – 4) and (1, – 2) is divided by y-axis in the ratio.

If the vertices of a triangle ABC are A ( – 4, – 1), B (1, 2) and C (4, – 3), then the coordinates of the circumcentre of the triangle are,

The mid points of the sides AB and AC of a triangle ABC are (2, – 1) and (– 4, 7) respectively, then the length of BC is

If O is the origin and the coordinates of A and B are ( x 1 , y 1 ) and ( x 2 , y 2 ) respectively then O A × O B cos ∠ A O B is equal to

If the lines x + a y + a = 0 , b x + y + b = 0 and c x + c y + 1 = 0 ( a , b , c being distinct and ≠ 1 ) are concurrent, then the value of a a − 1 + b b − 1 + c c − 1 is

If a, b, c are in A.P., then a x + b y + c = 0 represents

qx + py + ( p + q − r ) = 0 is the reflection of the line px + qy + r =0 in the line

If u = a 1 x + b 1 y + c 1 , v = a 2 x + b 2 y + c 2 = 0 and a 1 a 2 = b 1 b 2 ≠ c 1 c 2 then u + k v = 0

Reflection of the line x + y + 1 = 0 in the line l x + m y + n = 0 is

Area of the rhombus enclosed by the lines a x ± b y ± c = 0 is

The area enclosed by 2 | x | + 3 | y | ≤ 6 is

Two vertices of a triangle are (3, – 2) and (–2, 3) and its orthocenter is (–6, 1). Then the third vertex of this triangle can NOT lie on the line

If a line joining two points A (2, 0) and B (3, 1) is rotated about A in anticlockwise direction through an angle 15º, then equation of the line in the new position is

The orthocentre of the triangle formed by the lines y = 0 , ( 1 + t ) x − t y + t ( 1 + t ) = 0 and ( 1 + u ) x − u y + u ( 1 + u ) = 0 ( t ≠ u ) for all values of t and u lies on the line.

If the medians AD and BE of the triangle with vertices A ( 0 , b ) , B ( 0 , 0 ) and C ( a , 0 ) are perpendicular, then an equation of the line through (a, b) perpendicular to AC is

Let P = ( – 1 , 0 ) , Q = ( 0 , 0 ) and R = ( 3 , 3 3 ) be three points. Then the equation of the bisector of the angle P Q R is

If the sum of the distances of a point from two perpendicular lines is 1 then its locus is

The in centre of the triangle with vertices ( 1 , 3 ) , ( 0 , 0 ) and (2,0) is

If the points O ( 0 , 0 ) , A ( cos ⁡ α , sin ⁡ α ) , B ( cos ⁡ β , sin ⁡ β ) are the vertices of a right-angled triangle, then sin ⁡ α − β 2 =

if the points A ( λ , 2 λ ) , B ( 3 λ , 3 λ ) and C ( 3 , 1 ) are collinear, then λ =

If the foot of the perpendicular from the origin to straight line is at the point ( 3 , – 4 ) . Then the equation of the line is

A triangle with vertices ( 4 , 0 ) , ( − 1 , − 1 ) ( 3 , 5 ) , is

Point P ( 2 , 4 ) is translated through a distance 3 2 units measured parallel to the line y – x – 1 = 0 , in the direction of decreasing ordinates, to reach at Q . I f R is the image ofQ unth respect to the line y – x – 1 = 0 , then coordinates of R are

The angle through which the coordinates axes rotated so that xy-term in the equation 5 x 2 + 4 3 x y + 9 y 2 = 0 may be missing, is

The line ( p + 2 q ) x + ( p − 3 q ) y = p − q .for different values of p and q passes through the fixed point

Let k be an integer such that the triangle with vertices ( k , − 3 k ) , ( 5 , k ) and ( − k , 2 ) has area 28 sq. units. Then the orthocentre of this triangle is at the point:

One vertex of the equilateral triangle with centroid at the origin and one side as x + y – 2 = 0 , is

If the area of the parallelogram formed b:1 the lines 2 x − 3 y + a = 0 , 3 x − 2 y − a = 0 , 2 x − 3 y + 3 a = 0 and 3 x − 2 y − 2 a = 0 is 10 square units, then | a | =

If each of the points ( x 1 , 4 ) and ( – 2 , y 1 ) lies on the line joining the points ( 2 , – 1 ) and ( 5 , – 3 ) , then the point P ( x 1 , y 1 ) lies on the line

Point Q is symmetric to P (4, -1) with respect to the bisector of the first quadrant. If the length of P Q is k 2 , then k =

If ( a , a 2 ) falls inside the angle made by the lines y = x 2 , x > 0 and y = 3 x , x > 0 , , then a belongs to the interval

At what point the origin be shifted, if the, coordinates of a point (4, 5) become (-3, 9)?

The area of the triangle formed by the lines y = x , y = 0 and x = sin − 1 a 4 + 1 + cos − 1 ⁡ a 4 + 1 − tan − 1 ⁡ a 4 + 1 is sq. units, where k =

The distance between the lines 4 x + 3 y = 11 and 8 x + 6 y = 15 , is

The co-ordinate axes are rotated about the origin O in the counter-clockwise direction through an angle 60° If p and q are the intercepts made on the new axes by a straight line whose equation referred to original axes is x + y = 1 , then 1 p 2 + 1 q 2 =

The equation of one side of a rectangle is 3 x − 4 y − 10 = 0 and the coordinates of two of its vertices are ( − 2 , 1 ) and ( 2 , 4 ) .Then, the area of the rectangle, in square units, is

Orthocentre of the triangle with vertices ( 0 , 0 ) , ( 3 , 4 ) and ( 4 , 0 ) is

line through the point A ( 2 , 4 ) intersects the line x + y = 9 at the point P . The minimum distance of A P , is

The number of integer oalues of m, for which the x-coordinaie of the point of intersection of the lines 3 x + 4 y = 9 and y = m x + 1 is also an integer

The area ( in square units) enclosed by 2 | x | + 3 | y | ≤ 6 is

If the circumcentre of the triangle formed by the points ( 0 , 0 ) , ( 2 , − 1 ) and ( − 1 , 3 ) is at 5 2 , 5 2 then the coordinates of its orthcentre are

The distance between the lines 5 x − 12 y + 65 = 0 and 5 x − 12 y − 39 = 0 , is

The set of all possible values of θ in ( 0 , π ) for which the points P ( 1 , 2 ) and Q ( sin θ , cos θ ) lie on the same side of the lines x + y = 1 , is

If the points A ( − 2 , − 5 ) , B ( 2 , − 2 ) and C ( 8 , a ) are collinear. then a =

Point (1 ,2) and ( – 2 , 1 ) are

The equation of a line winch is equidistant from the lines x = − 4 and x = 8 , is

If Δ 1 is the area of the triangle formed by the centroid and two vertices of a triangle, Δ 2 is the area of the triangle formed by the mid-points of the sides of the same triangle, then Δ 1 : Δ 2 =

The distance of the line x + y – 8 = 0 from ( 4 , 1 ) measured along the direction whose slope is – 2 , is

The distance of the point ( 2 , 3 ) from the line 2 x – 3 y + 9 = 0 measured along the line x – y + 1 = 0 , is

The distance between the pair of lines given by x 2 + y 2 + 2 x y − 8 a x − 8 a y − 9 a 2 = 0 , is

The point (2, l) is shifted by 3 2 units parallel x + y = 1 the line in the direction of increasing ordinate, to reach Q The image of C2 in the line x + y = 1 , is

A line forms a triangle of area 54 3 sq-units with the coordinate axes. If the perpendicular drawn from the origin to the line makes an angle of 60 ° with the x – axis, then the equation the line is

The locus of the mid-point of the portion intercepted between the axes by the line x cos a + y sin a = p , where p is a constant, is

P ( 1 , 2 ) , Q ( 4 , 6 ) , R ( 5 , 7 ) and S ( a , b ) are the vertices of a parallelogram PQRS, then

Vertices of a variable triangle are A ( 3 , 4 ) , B ( 5 cos ⁡ θ , 5 sin ⁡ θ ) and C ( 5 sin ⁡ θ , − 5 cos ⁡ θ ) Then, locus of its orihocenire is

If two vertices of an equilateral triangle have integral coordinates, then the third vertex will have

The locus of the point of intersection of lines x cos ⁡ α + y sin ⁡ α = a and x sin ⁡ α − y cos ⁡ α = b ( α is a variable), is

If the pair of lines x 2 + 2 x y + a y 2 = 0 and a x 2 + 2 x y + y 2 = 0 have exactly one line in common, then a =

The locus of the mid-point of the portion intercepted between the axes by the line x cos a + y sin a = p , where p is a constant, is

The point P ( a , b ) lies on the straight line 3 x + 2 y = 13 and the point Q ( b , a ) lies on the straight line 4 x – y = 51 then the equation of the line P Q is

If the axes are shifted to the point (1, -2) without rotation, what does the equation 2 x 2 + y 2 − 4 x + 4 y = 0 become?

Let P = ( − 1 , 0 ) , Q = ( 0 , 0 ) and R = ( 3 , 3 3 ) be three points. Then, the equation of the bisector of the angle P Q R is