A light ray reflected from x = − 2 . If the reflected ray touches the circle x 2 + y 2 = 4 and the point of incident is ( − 2 , − 4 ) , then the equation of incedents ray
A circle touches the y-axis at the point (0, 4) and passes through the point(2, 0). Which of the following lines is not a tangent to this circle?
A B is a chord of the circle x 2 + y 2 = 25 . The tangents to the circle at A and B intersect at C . If ( 2 , 3 ) is the midpoint of A B , then the area of quadrilateral O A C B (Where O is origin) is
A circle is drawn touching the x-axis and centre at the point which is the reflection of ( a , b ) in the line y − x = 0 . The equation of the circle is
If the circumference of the circle x 2 + y 2 + 8 x + 8 y − b = 0 is bisected by the circle x 2 + y 2 − 2 x + 4 y + a = 0 then a + b =
If 3 x + y = 0 is a tangent to the circle with centre at the point (2, -1), then the equation of the other tangent to the circle from the origin, is
If the points (2, 0), (0, 1), (4, 5) and (0, c) are concyclic, then the value of 3c, is
The locus of the middle point of chords of the circle x 2 + y 2 = a 2 which pass through the fixed point ( h , k ) , is
Equation of a circle with centre ( 4 , 3 ) touching the circle x 2 + y 2 = 1 , is
If the lines 2 x + 3 y + 1 = 0 and 3 x − y − 4 = 0 lie along diameters of a circle of circumference 10 π , then the equation of the circle is
The tangent at P, any point on the circle x 2 + y 2 = 4 , meets the coordinate axes in A and B , then
The points of contact of tangents to the circle x 2 + y 2 = 25 which are inclined at an angle of 30 ° to the x – axis are
The pole of the straight line 9 x + y − 28 = 0 with respect to the circle 2 x 2 + 2 y 2 − 3 x + 5 y − 7 = 0 , is
The locus of the centre of a circle of radius 2 which rolls on the outside of the circle x 2 + y 2 + 3 x − 6 y − 9 = 0 , is
The equation a x 2 + 2 h x y + b y 2 + 2 g x + 2 f y + c = 0 represents a circle, the condition will be
The equation a x 2 + 2 h x y + b y 2 + 2 g x + 2 f y + c = 0 represents a circle, the condition will be
One of the diameters of the circle circumscribing the rectangle A B C D is 4 y = x + 7 . If A and Bare the points (-3,4) and (5, 4) respectively, then the area of the rectangle, in square units, is
The tangent to x 2 + y 2 = 9 which is parallel to y – axis and does not lie in the third quadrant touches the circle at the point
Extremities of a diagonal of a rectangle are ( 0, 0) and ( 4, 3). The equations of the tangents to the circumcircle of the rectangle which are parallel to the diagonal, are
Equation of the circle through the origin and making intercepts of 3 and 4 on the positive sides of the axes is
The equation of a circle which passes through ( 2 a , 0 ) and whose radical axis in relation to the circle x 2 + y 2 = a 2 is x = a / 2 , is
The coordinates of the point on the circle x 2 + y 2 − 12 x − 4 y + 30 = 0 , which is farthest from the origin, are
The area of the circle centred at (1, 2) and passing through (4, 6), is
A B is a diameter of a circle and C is any point on the circumference of the circle. Then,
The equation of a circle passing through ( 1 , 1 ) and points of intersection of x 2 + y 2 + 13 x − 3 y = 0 and 2 x 2 + 2 y 2 + 4 x − 7 y − 25 = 0 is
Equation of the circle which touches 3 x + 4 y = 7 and passes through ( 1 , − 2 ) and ( 4 , − 3 ) , is
The length of the tangent drawn from any point on the circle x 2 + y 2 + 2 g x + 2 f y + c = 0 to the circle x 2 + y 2 + 2 g x + 2 f y + c 1 = 0 is
The circle x 2 + y 2 = 4 cuts the circle x 2 + y 2 − 2 x − 4 = 0 at the points A and B. If the circle x 2 + y 2 − 4 x − k = 0 passes through A and B then the value of k , is
The circle x 2 + y 2 + 4 x − 7 y + 12 = 0 cuts an intercept on y-axis of length
If the chord x cos α + y sin α − p = 0 of the circle x 2 + y 2 − a 2 = 0 subtends a right angle at the centre of the circle, then a 2 p 2 is equal to
If g 2 + f 2 = c , then the equation x 2 + y 2 + 2 g x + 2 f y + c = 0 will represent
If ( m ; , 1 / m ; ) , i = 1 , 2 , 3 , 4 are concyclic points, then the value of m 1 m 2 m 3 m 4 , is
The equations of the circles which touch both the axes and the line x = a are
The greatest distance of the point P ( 10 , 7 ) from the circle x 2 + y 2 − 4 x − 2 y − 20 = 0 , is
if the distances from the origin of the centres of three circles x 2 + y 2 + 2 λ i x − c 2 = 0 ( i = 1 , 2 , 3 ) are in GP, then the lengths of the tangents drawn to them from any point on the circle x 2 + y 2 = c 2 are in
The equation of the circle passing through the point ( – 1 , 2 ) and having two diameters along the pair of lines x 2 − y 2 − 4 x + 2 y + 3 = 0 , is
If 2 x + 3 y − 6 = 0 and 9 x + 6 y − 18 = 0 cuts the axes in concyclic points, then the centre of the circle, is
The degree measure of the angle between x 2 + y 2 − 2 x − 2 y + 1 = 0 and line y = λ x + 1 − λ , is
The equation of the circle which is touched by y = x , has its centre on the positive direction of the x – a x i s and cuts off a chord of length 2 units along the line 3 y − x = 0 , is
Given that the circles x 2 + y 2 − 2 x + 6 y + 6 = 0 and x 2 + y 2 − 5 x + 6 y + 15 = 0 touch, the equation to their common tangent, is
The inverse point of ( 1 , – 1 ) with respect to x 2 + y 2 = 4 , is
The circle x 2 + y 2 = 4 cuts the circle x 2 + y 2 + 2 x + 3 y − 5 = 0 in A and B. Then the equation of the circle on AB as diameter, is
The equation of the circle passing through (0, 0) and belonging to the system of circles of which (3, 1) and (-1, 5) are limiting points, is
The equation x 2 − a 2 2 + y 2 − b 2 2 = 0 represents points
Circle ( s ) touching x – axis at a distance 3 from the origin and having an intercept of length 2 7 on y – axis, is (are)
The image of the circle x 2 + y 2 + 16 x − 24 y + 183 = 0 in the line mirror 4 x + 7 y + 13 = 0 , is
The equation of any tangent to the circle x 2 + y 2 − 2 x + 4 y − 4 = 0 is
The tangents drawn from the origin to the circle x 2 + y 2 − 2 r x − 2 h y + h 2 = 0 are perpendicular, if
The locus of the centre of a circle which cuts orthogonally the circle x 2 + y 2 − 20 x + 4 = 0 and which touches x = 2 is
The range of g so that we have always a chord of contact of tangents drawn from a real point ( a , a ) to the circle x 2 + y 2 + 2 g x + 4 y + 2 = 0 , is
The equation of the chord of the circle x 2 + y 2 − 6 x + 8 y = 0 0 which is bisected at the point ( 5 , − 3 ) is
If the area of the circle 4 x 2 + 4 y 2 − 8 x + 16 y + k = 0 is 9 π square units, then the value of k is
The line 3 x – 2 y = k meets the circle x 2 + y 2 = 4 r 2 at only one point, if k 2 =
The radical axis of two circles having centres at C1 and C 2 and radii r1 and r2 is neither intersecting nor touching any of the circles, if
The number of common tangents of the circles ( x + 3 ) 2 + ( y – 2 ) 2 = 49 and ( x – 2 ) 2 + ( y + 1 ) 2 = 4 is
The locus of the center of a circle which cuts orthogonally the circle x 2 + y 2 − 20 x + 4 = 0 and which touches x = 2 , is
If real numbers x and y satisfy (x+5) 2 + ( y – 12 ) 2 = 14 2 , t h e n t h e m i n i m u m v a l u e o f x 2 + y 2 i s
The tangent at any point to the circle x 2 + y 2 = r 2 meets the coordinate axes at A and B. If the lines drawn parallel to the coordinate axes through A & B intersect at P. Then the locus of P is
A circle touches the x – axis and also touches the circle with centre at ( 0 , 3 ) and radius 2 . The locus of the centre of the circle is
The tangents drawn from the origin to the circle x 2 + y 2 – 2 p x – 2 q y + q 2 = 0 are perpendicular if
For all values of a, equation of a diameter of the circle x 2 + y 2 − 2 a x + 2 a y + a 2 = 0 is
S : x 2 + y 2 − 6 x + 4 y − 3 = 0 is a circle and L : 4 x + 3 y + 19 = 0 is a straight line.
If a circle passes through ( a , b ) and cuts the circle x 2 + y 2 = 4 orthogonally, then the locus of its centre is
If a chord of a circle x 2 + y 2 = 25 with one extremity at ( 4 , 3 ) subtends a right angle at the centre of this circle, then the coordinates of the other extremity of this chord can be
The circle passing through the point ( – 1 , 0 ) and touching y – axis at ( 0 , 2 ) also passes through the point
Let A B be a chord of the circle x 2 + y 2 = r 2 subtending a right angle at the centre. Then the locus of the centroid of the triangle P A B as P moves on the circle is
The lines 2 x − 3 y − 5 = 0 and 3 x − 4 y = 7 are diameters of a circle of area 154 ( = 49 π ) sq. units, then the equation of the circle is
If a circle passes through ( 1 , 2 ) and cuts the circle x 2 + y 2 = 4 orthogonally then the equation of the locus of its centre, is
The area of the triangle, in square units, formed by the tangents from the point ( 4 , 3 ) to the circle x 2 + y 2 = 9 and the line joining their points of contact, is
A square is inscribed in the circle x 2 + y 2 − 2 x + 4 y − 93 = 0 with its sides parallel to the coordinate axes. The coordinates of its vertices, are
The locus of a point represented by x = a 2 t + 1 t , y = a 2 t − 1 t , is
If 2 x − 4 y = 9 and 6 x − 12 y + 7 = 0 are parallel tangents to circle, then radius of the circle, is
The equations of the tangents drawn from (7, 1) to the circle x 2 + y 2 = 25 are
The equation of the circle which has a tangent 2 x − y − 1 = 0 at (3,5) on it and with the centre on x + y = 5 is
The number of integral values of λ for which the equation x 2 + y 2 − 2 λ x + 2 λ y + 14 = 0 represents a circle whose radius cannot exceed 6 , is
The normal at the point (3, 4) on a circle at the point (-1,-2). The equation of the circle, is
If the circles ( x − a ) 2 + ( y − b ) 2 = c 2 and ( x − b ) 2 + ( y − a ) 2 = c 2 touch each other, then
The equation a x 2 + 2 h x y + b y 2 + 2 g x + 2 f y + c = 0 represents a circle, the condition will be
The equation of the circle which touches the axes of the coordinates and the line x 3 + y 4 = 1 and whose centre lies in the first quadrant is x 2 + y 2 – 2 c x – 2 c y + c 2 = 0 , where c , is
The centre of the circle inscribed in the square formed by the lines x 2 − 8 x + 12 = 0 and y 2 − 14 y + 45 = 0 , is
The equation of a circle which cuts the three circles x 2 + y 2 − 3 x − 6 y + 14 = 0 , x 2 + y 2 − x − 4 y + 8 = 0 x 2 + y 2 + 2 x − 6 y + 9 = 0 orthogonally, is
The number of common tangents that can be drawn to the circle x 2 + y 2 − 4 x − 6 y − 3 = 0 and x 2 + y 2 + 2 x + 2 y + 1 = 0 is
The two circles x 2 + y 2 − 2 x − 3 = 0 and x 2 + y 2 − 4 x − 6 y − 8 = 0 are such that
The circles x 2 + y 2 + x + y = 0 and x 2 + y 2 + x − y = 0 intersect at an angle of
If the chord of contact of tangents drawn from the point (h, k) to the circle x 2 + y 2 = a 2 subtends a right angle at the centre, then
If the straight line x − 2 y + 1 = 0 intersects the circle x 2 + y 2 = 25 in points P and Q , then the coordinates of the point of intersection of tangents drawn at P and Q to the circle x 2 + y 2 = 25 are
The number of common tangents of the circles x 2 + y 2 − 2 x − 1 = 0 and x 2 + y 2 − 2 y − 7 = 0 , is
f the tangent at the point P on the circle x 2 + y 2 + 6 x + 6 y = 2 meets the straight line 5 x − 2 y + 6 = 0 at a point Q on the y-axis, then the length of P Q , is
The equation of the circle whose one diameter is PQ, where the ordinates of P, Q are the roots of the equation x 2 + 2 x − 3 = 0 and the abscissae are the roots of the equation y 2 + 4 y − 12 = 0 ,is
One of the limit point of the coaxial system of circles containing x 2 + y 2 − 6 x − 6 y + 4 = 0 , x 2 + y 2 − 2 x − 4 y + 3 = 0 , is
If the circle x 2 + y 2 + 2 g x + 2 f y + c = 0 is touched by y = x at P such that O P = 6 2 , then the value of c is
The set of values of ‘ a ‘ for which the point ( a − 1 , a + 1 ) lies outside the circle X 2 + Y 2 = 8 and inside the circle x 2 + y 2 − 12 x + 12 y − 62 = 0 , is
Two perpendicular tangents to the circle x 2 + y 2 = a 2 meet at P . Then the locus of P has the equation
Two perpendicular tangents to the circle x 2 + y 2 = a 2 meet at P . Then the locus of P has the equation
The coordinates of the middle point of the chord intercepted on line l x + m y + n = 0 by the circle x 2 + y 2 = a 2 are
The straight line x a + y b = 1 cuts the coordinate axes at A and B . The equation of the circle passing through O ( 0, 0), A and B , is
ABCD is a square whose side is a. The equation of the circle circumscribing the square, taking AB and AD as axes of reference, is
The equation of the circle having its centre on the line x – 2 y – 3 = 0 and passing through the point of intersection of the circles x 2 + y 2 − 2 x − 4 y + 1 = 0 and x 2 + y 2 − 4 x − 2 y + 4 = 0 is
How many common tangents can be drawn to the following circles x 2 + y 2 = 6 x and x 2 + y 2 + 6 x + 2 y + 1 = 0 ?
The length of the chord of the circle x 2 + y 2 + 4 x − 7 y + 2 = 0 along the y-axis, is
There are two circles whose equations are x 2 + y 2 = 9 and x 2 + y 2 − 8 x − 6 y + n 2 = 0 , n ∈ Z having exactly two common tangents. The number of possible values of n , is
The set of values of a for which the point ( 2 a , a + 1 ) is an interior point of the larger segment of the circle x 2 + y 2 – 2 x – 2 y – 8 = 0 made by the chord x – y + 1 = 0 , is
The equations of the tangents drawn from the origin to the circle x 2 + y 2 − 2 r x − 2 h y + h 2 = 0 , is
If the line y = m x – ( m – 1 ) cuts the circle x 2 + y 2 = 4 at two real and distinct points, then
If P ( 1 , 1 / 2 ) is a centre of similitude for the circles x 2 + y 2 + 4 x + 2 y − 4 = 0 and x 2 + y 2 − 4 x − 2 y + 4 = 0 , then the length of the common tangent through P to the circles, i
The circles x 2 + y 2 − 10 x + 16 = 0 and x 2 + y 2 = r 2 intersect each other in two distinct points if
x , y ∈ R satisfies ( x + 5 ) 2 + ( y – 12 ) 2 = 14 2 , then the min value of x 2 + y 2
Let the tangents drawn from the origin to the circle, x 2 + y 2 − 8 x − 4 y + 16 = 0 touch it at the points A and B. Then A B 2 is equal to:
There are two circles whose equations are x2+y2=9 and x 2 + y 2 – 8 x – 6 y + n 2 = 0 , n ∈ I . If the two circles having exactly two common tangents, then the number of the possible values of ‘n’ is
Let AB be a chord of the circle x 2 + y 2 = r 2 subtending a right angle at the centre. Thus the locus of the centroid of the triangle PAB as P moves on the circle is
Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals
Consider a family of circles which are passing through M 1 , 1 and are tangent to x-axis. If h , k is the centre of circle, then:
Number of values of c for which set ( x , y ) : x 2 + y 2 + 2 x ≤ 1 ∩ { ( x , y ) : x − y + c ≥ 0 } contains only one element
The square of diameter of the circle having tangent at ( 1 , 1 ) as x + y − 2 = 0 and passing through (2, 2) is ……
Let A B be the chord of contact of the point ( 5 , − 5 ) with respect to the circle x 2 + y 2 = 5 . Then the locus of the orthocenter of Δ P A B , where P is any point on the circle, is
The tangent and normal at the point A ( 4 , 4 ) to the parabola y 2 = 4 x intersect x -axis at the points B and C respectively, then the equation of circumcircle of △ ABC is
Equation of the circle with centre on the y-axis and passing through the origin and the point (2, 3) is
The sum of all integral value(s) of ‘ r ‘ for which the circles x 2 + y 2 − 10 x + 16 y + 89 − r 2 = 0 and x 2 + y 2 + 6 x − 14 y + 42 = 0 intersect in two real distinct points is
If the point (3,4) lies inside and the point (-3,-4) lies outside the circle x 2 +y 2 -7x+ 5y -p =0. then the set of all possible values of p is
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 locus of the center of the circles such that the point (2,3) is the midpoint of the chord 5x + 2y= 16 is
A circle of constant radius a passes through the origin 0 and cuts the axes of coordinates at points P and Q. Then the equation of the locus of the foot of perpendicular from O to PQ is
If the chord y = mx + 1 of the circles x 2 + y 2 = 1 subtends an angle of 45° at the major segment of the circle, then the value of m is
(-6,0), (0, 6), and (-7,7) are the vertices of a ∆ ABC .The incircle of the triangle has equation
The equations of four circles are ( x ± a ) 2 + ( y ± a ) 2 = a 2 The radius of a circle touching all the four circles is
A region in the x-y plane is bounded by the curve y = 25 − x 2 and the line y =0 If the point (a, a+ 1) lies in the interior of the region then,
The range of values of λ , λ > 0 such that the angle θ between the pair of tangents drawn from ( λ , 0 ) to the circle x 2 + y 2 = 4 lies in ( π / 2 , 2 π / 3 ) is
Let P be any moving point on the circle x 2 + y 2 − 2 x = 1 AB be the chord of contact of this point w.r.t. the circle x 2 + y 2 − 2 x = 0 The locus of the circumcenter of triangle CAB (C being the center of the circle) is
If lines 2x – 3y + 6 = 0 and kx + 2y + 12 =0 cut the co-ordinate axes in concyclic points then the value of k is
The equation of the chord of the circle x 2 + y 2 − 3 x − 4 y − 4 = 0 which passes through the origin such that the origin divides it in the ratio 4 : 1, is
If a circle of constant radius 3k passes through the origin O and meets the coordinate axes at A and B, then the locus of the centroid of triangle OAB is
Equations of the common tangents of the circles x 2 + y 2 – 2 x – 6 y + 9 = 0 and x 2 + y 2 + 6 x – 2 y + 1 = 0 are A) x = 0 B) y = 4 C) 3x +4 y =10
A circle whose radius is 5 and which touches externally the circle x 2 + y 2 – 2 x – 4 y – 20 = 0 at the point ( 5 , 5 ) intersects in real distinct points the line
Equation of a circle with centre ( – 4 , 3 ) touching internally and containing the circle x 2 + y 2 = 1 is
Circle x 2 + y 2 + 2 x − 8 y − 8 = 0 and x 2 + y 2 + 2 x − 6 y − 6 = 0
Vertices of an isosceles triangle of area a 2 and ( − a , 0 ) and ( a , 0 ) . Equation of the circumcircle of the triangle is
A circle is described on the line joining the points ( 2 , − 3 ) and ( − 4 , 7 ) as a diameter
Consider a family of circles which are passing through the point ( – 1 , 1 ) and are tangent to x – axis. If ( h , k ) are the coordinates of the centre of the circles, then the set of values of k is given by the interval
The centre of the circle passing through (0, 0) and (1, 0) and touching the circle x 2 + y 2 = 9 ,is
If the lines 3 x – 4 y – 7 = 0 and 2 x – 3 y – 5 = 0 are diameters of a circle of area 49 π square units, the equation of the circle, is
A variable circle passes through the fixed point (2, 0) and touches the y-axis. Then, the locus of its centre, is
If a point ( α , β ) lies on the circle x 2 + y 2 = 1 then the locus of the point ( 3a + 2, P), ( 3 α + 2 , β ) , is
The number of real tangents that can be drawn from ( 2 , 2 ) to the circle x 2 + y 2 − 6 x − 4 y + 3 = 0 is
if 2 x 2 + y 2 + 4 λ x + λ 2 = 0 represents a circle of meaningful radius, then the range of real values of λ ., is
If the base of a triangle and the ratio of the lengths of the other two unequal sides are given, then the vertex lies on a / a n
The length of the tangent from (0, 0) to the circle 2 x 2 + y 2 + x − y + 5 = 0 is
Tangents are drawn from a point on the circle x 2 + y 2 − 4 x + 6 y − 37 = 0 to the circle x 2 + y 2 − 4 x + 6 y − 12 = 0 The angle between the tangents, is
The lengths of the tangents from any point on the circle 15 x 2 + 15 y 2 − 48 x + 64 y = 0 to the circles 5 x 2 + 5 y 2 − 24 x + 32 y + 75 = 0 5 x 2 + 5 y 2 − 48 x + 64 y + 300 = 0 are in the ratio
The angle between two tangents drawn from the origin to the circle ( x − 7 ) 2 + ( y + 1 ) 2 = 25 , is
Equation of a circle which passes through (3, 6) and touches the axes, is
The straight line x − 2 y + 1 = 0 intersects the Circle x 2 + y 2 = 25 in points P and Q. The coordinates of the point of intersection of tangents drawn at P and Q to the circle are
Two circles, each of radius 5, have a common tangent at (1, 1) whose equation is 3 x + 4 y – 7 = 0 . Then, their centres, are
The locus of a point which moves such that the tangents from it to the two circles x 2 + y 2 − 5 x − 3 = 0 and 3 x 2 + 3 y 2 + 2 x + 4 y − 6 = 0 are equal, is given by
The angle between the tangents drawn from a point on the director circle e x 2 + y 2 = 50 to the circle x 2 + y 2 = 25 is
If the line l x + m y + n = 0 intersects the curve a x 2 + 2 h x y + b y 2 = 1 at P and Q such that the circle with PQ as a diameter passes through the origin, then l 2 + m 2 =
The two conics a 1 x 2 + 2 h 1 x y + b 1 y 2 = c 1 and a 2 x 2 + 2 h 2 x y + b 2 y 2 = c 2 intersect in 4 concyclic points. Then
If the point ( a , – a ) lies inside the circle x 2 + y 2 − 4 x + 2 y − 8 = 0 , then a lies in the interval
For the given circles 5 x 2 + y 2 − 6 x − 2 y + 1 = 0 and x 2 + y 2 + 2 x − 8 y + 13 = 0 which of the following is true?
The coordinates of the middle point of the chord intercepted on line I x + m y + n = 0 by the circle x 2 + y 2 = a 2 are
The equation of the chord of the circle x 2 + y 2 − 6 x + 8 y = 0 which is bisected at the point ( 5 , − 3 ) , IS
The geometric mean of the minimum and maximum values of the distance of point ( – 7 , 2 ) , from the points on the circle x 2 + y 2 − 10 x − 14 y − 51 = 0 is equal to m n , where m , n are coprimes, then m n =
If 3 x 2 + 2 λ x y + 3 y 2 + ( 6 − λ ) x + ( 2 λ − 6 ) y − 21 = 0 is the equation of a circle, then its radius is
The equation λ 2 x 2 + λ 2 − 5 λ + 4 x y + ( 3 λ − 2 ) y 2 − 8 x + 12 y − 4 = 0 will represent a circle, if λ =
If ( 2 , 4 ) is a point interior to the circle x 2 + y 2 − 6 x − 10 y + λ = 0 and the circle does not cut the axes at any point, then
The number of common tangents to the circles x 2 + y 2 − 4 x − 6 y − 12 = 0 and x 2 + y 2 + 6 x + 18 y + 26 = 0 , is
The locus of the centre of the circle which cuts the circles x 2 + y 2 + 2 g 1 x + 2 f 1 y + c 1 = 0 and x 2 + y 2 + 2 g 2 x + 2 f 2 y + c 2 = 0 orthogonally, is
The coordinates of the centre and radius of the circle represented by the equation ( 3 − 2 λ ) x 2 + λ y 2 − 4 x + 2 y − 4 = 0 are
PQ is a chord of the circle x 2 + y 2 − 2 x − 8 = 0 whose midpoint is (2, 2). The circle passing through P, Q and (1, 2) is
A , B , C and D are the points of intersection with the coordinate axes of the lines a x + b y = a b and b x + a y = a b then ,
The point diametrically opposite to the point P ( 1 , 0 ) on the circle x 2 + y 2 + 2 x + 4 y − 3 = 0 , is
The locus of the centre of the circle passing through the origin O and the points of intersection A and B of any line through ( a , b ) and the coordinate axes is a x + b y = λ ,where λ
For the two circles x 2 + y 2 = 16 and x 2 + y 2 − 2 y = 0 there is/are
The centre of the circle x = 2 + 3 cos θ , y = 3 sin θ – 1 , is
The locus of the centre of a circle which touches externally the circle x 2 + y 2 − 6 x − 6 y + 14 = 0 and also touches the y-axis is given by the equation
Two equal circles with their centres on x and y axis will possess the radical axis in the following form
There are two circles whose equations are x 2 + y 2 = 9 and x 2 + y 2 − 8 x − 6 y + n 2 = 0 , n ∈ Z , having exactly two common tangents. The number of possible values of n , is
The locus of the centre of a circle touching the lines x + 2 y = 0 and x – 2 y = 0 , is
The locus of the middle points of chords of the circle x 2 + y 2 = 25 which are parallel to the line x − 2 y + 3 = 0 is
The equation of the circle on the common chord of the circles ( x − a ) 2 + y 2 = a 2 and x 2 + ( y + b ) 2 = b 2 as diameter is ,
Let there be n ( ≥ 3 ) circles. The value of n for which the number of radical axis of these circles is same as the number of radical centres, is
If the points A (2, 5) and B are symmetrical about the tangent to the circle x 2 + y 2 − 4 x + 4 y = 0 at the origin then the coordinates of B , are
Circles are drawn through the point ( 3 , 0 ) to cut an intercept of length 6 units on the negative direction of the x – a x i s . The equation of the locus of their centres, is
Two tangents to the circle x 2 + y 2 = 4 at the points A and B meet at P (- 4, 0). The area of the quadrilateral P A O B , where O is the origin, is
The equation of the circumcircle of an equilateral triangle is x 2 + y 2 + 2 g x + 2 f y + c = 0 and one vertex of the triangle is ( 1 , 1 ) . The equation of the incircle of the triangle, is
If the length of the chord of the circle x 2 + y 2 = r 2 along the line y – 2 x = 3 is r , then r 2 is equal to
The degree measure of the angle between x 2 + y 2 − 2 x − 2 y + 1 = 0 and line y = λ x + 1 − λ , is
The intercept on the line y = x by the circle x 2 + y 2 − 2 x = 0 is AB. The equation of the circle with AB as diameter, is
α , β and y are the parametric angles of three points P , Q , and R respectively, on the circle x 2 + y 2 = 1 , and A is the point ( – 1 , 0 ) . If the lengths of the chords A P , A Q and A R are in G P , then cos α / 2 , cos β / 2 and cos γ / 2 are in
The radical axis of the circles x 2 + y 2 + 2 g x + 2 f y + c = 0 and 2 x 2 + y 2 + 3 x + 2 y + 2 c = 0 touches the circle x 2 + y 2 + 2 x + 2 y + 1 = 0 , if
The point from which the tangents to the circles x 2 + y 2 − 8 x + 40 = 0 5 x 2 + 5 y 2 − 25 x + 80 = 0 x 2 + y 2 − 8 x + 16 y + 160 = 0 are equal in length, is
The equation of the smallest circle passing from points ( 1 , 1 ) and ( 2 , 2 ) and always in the first quadrant, is
If 3 x + y = 0 tangent to the circle having its centre at (2, -1), then the equation of other tangent to the circle from the origin, is
A line is drawn through a fixed point P ( α , β ) to cut the circle x 2 + y 2 = r 2 at A and B .Then P A ⋅ P B is equal to
Four distinct points ( 2 k , 3 k ) , ( 1 , 0 ) , ( 0 , 1 ) and ( 0 , 0 ) lie on a circle for
A line meets the coordinate axes in A and B . A circle is circumscribed about the triangle O A B . The distances from the end points A , B of the side AB to the tangent at O are equal to m and n respectively. Then, the diameter of the circle, is
The equation of a circle passing through ( 3, – 6) and touching both the axes, is
The locus of the mid points of a chord of the circle x 2 + y 2 = 4 which subtends a right angle at the origin, is
The distance between the chords of contact of the tangent to the circle x 2 + y 2 + 2 g x + 2 f y + c = 0 from the origin and the point ( g , f ) is
Equation of the chord of the circle x 2 + y 2 − 4 x = 0 whose mid point is ( 1 , 0 ) , is
The tangents to x 2 + y 2 = a 2 having inclinations α and β intersect at P . If cot α + cot β = 0 then the locus of P is
The circles whose equations are x 2 + y 2 + c 2 = 2 a x and x 2 + y 2 + c 2 − 2 b y = 0 will touch one another externally, if
The locus of the centre of the circles which touch both the circles x 2 + y 2 = a 2 and x 2 + y 2 = 4 a x externally has the equation
The equation of the circle passing through ( 4 , 5 ) having the centre at ( 2 , 2 ) , is
The number of the tangents that can bedrawnfrom(l, 2) ( 1 , 2 ) to x 2 + y 2 = 5 is
The equation of the circle having radius 5 and touching the circle x 2 + y 2 − 2 x − 4 y − 20 = 0 at ( 5 , 5 ) is
The pole of a straight line with respect to the circle x 2 + y 2 = a 2 lies on the circle x 2 + y 2 = 9 a 2 If the straight line touches the circle x 2 + y 2 = r 2 then a 2 r 2 is equal to
The length of the common chord of the circles x 2 + y 2 + 4 x + 1 = 0 and x 2 + y 2 + 4 y − 1 = 0 , is
The length of the common chord of the circles x 2 + y 2 + 4 x + 1 = 0 and x 2 + y 2 + 4 y − 1 = 0 , is
If the equation x 2 + y 2 + 6 x − 2 y + λ 2 + 3 λ + 12 = 0 represents a circle, then
One of the diameters of the circle x 2 + y 2 − 12 x + 4 y + 6 = 0 is given by
The equation a x 2 + 2 h x y + b y 2 + 2 g x + 2 f y + c = 0 represents a circle, the condition will be
Let P be a point on the circle x 2 + y 2 = 9 , Q a point on the line 7 x + y + 3 = 0 , and the perpendicular bisector of P Q be the line x − y + 1 = 0 . Then, the coordinates of P are
The equation of the circle passing through ( 4, 3 having the centre (2, 2), is
If a circle has two of its diameters along the lines x + y = 5 and x – y = 1 and has area 9 π , then the equation of the circle is
The two circles x 2 + y 2 − 5 = 0 and x 2 + y 2 − 2 x − 4 y − 15 = 0
The point on the straight line y = 2 x + 11 which is nearest to the circle 16 x 2 + y 2 + 32 x − 8 y − 50 = 0 , is
If from the origin a chord is drawn to the circle x 2 + y 2 – 2 x = 0 , then the locus of the mid point of the chord has equation
The equation x 2 + y 2 − 6 x + 8 y + 25 = 0 represents
The slope of the tangent at the point ( h , h ) of the circle x 2 + y 2 = a 2 , is
The equation a x 2 + 2 h x y + b y 2 + 2 g x + 2 f y + c = 0 represents a circle, the condition will be
The coordinates of the middle point of the chord cut off by 2 x − 5 y + 18 = 0 on the circle x 2 + y 2 − 6 x + 2 y − 54 = 0 , are
The number of common tangents to the circles x 2 + y 2 − x = 0 , x 2 + y 2 + x = 0 , is
The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3 a is
If the centroid of an equilateral triangle . (2, – 2) and its one vertex is (-1, 1), then the equation of circumcircle is
The coordinates of the centre of the circle which intersects Circles x 2 + y 2 + 4 x + 7 = 0 , 2 x 2 + 2 y 2 + 3 x + 5 y + 9 = 0 and x 2 + y 2 + y = 0 orthogonally are
The angle of intersection of the circles x 2 + y 2 = 4 and x 2 + y 2 = 2 x + 2 y , is
If the circle x 2 + y 2 + 2 x + 3 y + 1 = 0 cuts x 2 + y 2 + 4 x + 3 y + 2 = 0 in A and B , then the equation of the circle on A B as diameter, is
If the chords of contact of the tangents from a point on the circle x 2 + y 2 = a 2 to the circle x 2 + y 2 = b 2 touch the circle x 2 + y 2 = c 2 , then the roots of the equation a x 2 + 2 b x + c = 0 , are
If the tangents are drawn to the circle x 2 + y 2 = 12 at the point where it meets the circle x 2 + y 2 − 5 x + 3 y − 2 = 0 then the point of intersection of these tangents, is
The lines 3 x – 4 y + 4 = 0 and 6 x – B y – 7 = 0 are tangents to the same circle. Then, its radius, is
The equation of the circle whose centre is (2, – 3) and radius 5, is
