MathsMaths QuestionsConic Section Questions for CBSE Class 11th

Conic Section Questions for CBSE Class 11th

A normal to the hyperbola x 2 6 − y 2 2 = 1 has equal intercepts on positive x and y axes. If this normal touches the ellipse x 2 a 2 + y 2 b 2 = 1 , then the value of a 2 + b 2 is

Ratio of the greatest and least focal distances of a point on the ellipse 4 x 2 + 9 y 2 = 36 is

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    If P S Q and P S ′ R are two chords of an ellipse through its foci S and S respectively, then P S S Q + P S ′ S ′ R =

    A set of parallel chords of the parabola y 2 = 4 a x have their midpoints on

    If the line x + 2 y + 4 = 0 is a common tangent to the parabolas y 2 = 4 a x and x 2 = 4 b y , then | b − a | is

    The locus of a point which divides the line segment joining the point ( 0 , – 1 ) and a point on the parabola x 2 = 4 y internally in the ratio 1 : 2 is:

    The equation of the parabola having focus at 6 , 0 a n d d i r e c t r i x a t x = – 6

    A tangent is drawn to the parabola y 2 = 4 x at the point P whose abscissa lies in the interval [ 1 , 4 ] . The maximum possible area of the triangle formed by the tangent at P , the ordinates of the point P , and the x -axis is equal to

    A B is a chord of the parabola y 2 = 4 a x with vertex A ⋅ B C is drawn perpendicular to A B meeting the axis at C . The projection of B C on the axis of the parabola is

    Two parabolas have the same focus. If their directrices are the x and the y-axis, respectively, then the slope of their common chord is

    Tangents are drawn to the parabola ( x − 3 ) 2 + ( x + 4 ) 2 = ( 3 x − 4 y − 6 ) 2 25 at the extremities of the chord 2 x − 3 y – 18 = 0 Then the angle between the tangents is

    Let y = 3 x − 8 be the equation of the tangent at the point (7, 13) lying on a parabola whose focus is at ( − 1 , − 1 ) . The length of the latus rectum of the parabola is

    An ellipse passing through the origin has its foci at ( 3 , 4 ) and ( 6 , 8 ) , and length of its semi-minor axis L . The value of L / 2 is

    Tangents drawn from the point ( a , 2 ) to the hyperbola x 2 16 − y 2 9 = 1 are perpendicular then the value of a 2 i s

    The equation of the ellipse whose axes are coincident with the coordinates axes and which touches the straight lines 3 x − 2 y − 20 = 0 and x + 6 y − 20 = 0 is

    The eccentric angle of a point on the ellipse x 2 4 + y 2 3 = 1 at a distance of 5 / 4 units from the focus on the positive x- axis is

    The eccentricity of the hyperbola x 2 a 2 − y 2 b 2 = 1 is reciprocal to that of the ellipse x 2 + 4 y 2 = 4 . If the hyperbola passes through a foci of the ellipse, then

    The absolute value of slope of common tangents to parabola y 2 = 8 x and hyperbola 3 x 2 − y 2 = 3 is

    Length of the latus rectum of the hyperbola x y = c 2 is equal to

    A point R divides the line segment joining the points P ( 1 , 3 ) and Q ( 1 , 1 ) in the ratio 1 : λ If R is an interior point of the parabola y 2 = 4 x , then λ can take a value in the interval

    If 3 x + 4 y = 12 2 is a tangent to the ellipse x 2 a 2 + y 2 9 = 1 for some a ∈ R , then the distance between the foci the of ellipse is

    The minimum distance of any point on the ellipse x 2 + 3 y 2 + 4 x y = 4 from its centre is

    Let P be any point on a directrix of an ellipse with eccentricity e. S be the corresponding focus and C the centre of the ellipse. The line PC meets the ellipse at A. The angle between PS and tangent at A is α . Then α is equal to

    If line y = x + 2 do not intersect any member of family of parabolas y 2 = α x a ∈ R + at two distinct points, then maximum value of latus rectum is

    The equation ( 5 x − 1 ) 2 + ( 5 y − 2 ) 2 = λ 2 − 2 λ + 1 ( 3 x + 4 y − 1 ) 2 represents an ellipse if λ ∈

    The minimum distance between the curves y 2 = 4 x and x 2 + y 2 − 12 x + 31 = 0 is

    The equation of the common tangent touching the circle ( x − 3 ) 2 + y 2 = 9 and the parabola y 2 = 4 x above the x- axis is

    If line y = x + 2 does not intersect any member of family of parabolas y 2 = a x , a ∈ R + at two distinct points, then the maximum length of latus rectum of parabola is

    If 2 x + y + λ = 0 is a normal to the parabola y 2 = − 8 x , then λ is

    Let the tangent at A ( p , q ) to the ellipse x 2 16 + y 2 4 = 1 form a triangle of least area with the coordinate axes ( p , q > 0 ) . Then p 2 + q 2 is equal to

    If a x 2 + y 2 + 2 y + 1 = ( x − 2 y + 3 ) 2 is an ellipse and a ∈ ( b , ∞ ) , then the value of b is

    A pair of tangents S A and S B are drawn to parabola y 2 = 8 x from a foci S of ellipse x 2 a 2 + y 2 b 2 = 1 (with a > b ), such that Δ S A B is an equilateral triangle. If the tangents S A and S B pass through the extremities of minor axis of ellipse, then 4 e 2 equals (where e is eccentricity of ellipse)

    Number of integral values of b for which tangent parallel to line y = x + 1 can be drawn to hyperbola x 2 5 − y 2 b 2 = 1 i s

    If distance between two parallel tangents having slope m drawn to the hyperbola x 2 9 − y 2 49 = 1 is 2 then the value of | m | is

    Let P x 1 , y 1 and Q x 2 , y 2 , y 1 < 0 , y 2 < 0 , be the endpoints of the latus rectum of the ellipse x 2 + 4 y 2 = 4 . The equations of parabolas with latus rectum P Q are

    The equation of the ellipse having focus ( 1 , − 1 ) , directrix x − y − 3 = 0 and eccentricity 1 / 2 is

    A circle has the same center as an ellipse and passes through the foci F 1 , and F 2 of the ellipse, such that the two curves intersect at four points. Let P be any one of their points of intersection. If the major axis of the ellipse is 17 and the area of triangle P F 1 F 2 is 30 , then the distance between the foci is

    Two circles are given such that one is completely lying inside the other without touching. Then the locus of the centre of a variable circle which touches the smaller circle from outside and the bigger circle from inside is

    If tangents are drawn to the ellipse x 2 + 2 y 2 = 2 , then the locus of the midpoint of the intercept made by the tangents between the coordinate axes is

    If the normal at the point P ( θ ) to the ellipse x 2 14 + y 5 = 1 intersects it again at the point Q ( 2 θ ) , then cos ⁡ θ is equal to

    If α + β = π then the chord joining the points α and β for the hyperbola x 2 a 2 − y 2 b 2 = 1 passes through

    If the eccentricity of the ellipse x 2 a 2 + 1 + y 2 a 2 + 2 = 1 is 1 / 6 , then the latus rectum of the ellipse is

    Let S and S ′ be two foci of the ellipse x 2 a 2 + y 2 b 2 = 1 . If a circle described on S S ′ as diameter intersects the ellipse at real and distinct points, then the eccentricity e of the ellipse satisfies

    The locus of the point which divides the double ordinates of the ellipse x 2 a 2 + y 2 b 2 = 1 in the ratio 1 : 2 internally is

    If (1,1) and (6, 1) are the vertices of an ellipse and one of whose foci lies on x – 2y = 2 then the eccentricity of ellipse is

    The eccentricity of the locus of point ( 3 h + 2 , k ) , where ( h , k ) lies on the circle x 2 + y 2 = 1 , is

    The minimum area of the triangle formed by the tangent to x 2 a 2 + y 2 b 2 = 1 and the coordinate axes is

    The slopes of the common tangents of the ellipse x 2 4 + y 2 1 = 1 and the circle x 2 + y 2 = 3 are

    If e is the eccentricity of the hyperbola x 2 a 2 − y 2 b 2 = 1 and θ is the angle between the asymptotes, then cos ⁡ θ 2 is equal to

    lf p, q are the segments of a focal chord of an ellipse b 2 x 2 + a 2 y 2 = a 2 b 2 , then

    A man running a race course notes that the sum of distance of two flag posts from him is always 10 meters and distance between the flag post’s is 8 meters and Locus of man is a conic then length of the latus rectum of the conic is

    Equation of the ellipse having centre at origin and touching the hyperbola x 2 25 − y 2 16 = 1 at its vertices and having eccentricity 3/5 is

    Equation of common tangent to y 2 = 8 x and x 2 + y 2 = 2 is

    Tangents are drawn to the parabola ( x − 3 ) 2 + ( x + 4 ) 2 = ( 3 x − 4 y − 6 ) 2 25 at the extremities of the chord 2 x − 3 y – 18 = 0 Then the angle between the tangents is

    The equation 16 x 2 − 3 y 2 − 32 x + 12 y − 44 = 0 represents a hyperbola

    The line x + y = 6 is a normal to the parabola. y 2 = 8 x at the point.

    Normal at a point P on the parabola y 2 = 4 a x meets the axis at Q such that the distance of Q from the focus of the parabola is 10 a . The coordinates of P are:

    The directrix of the parabola y 2 + 4 x + 3 = 0 is

    The sum of the squares of the perpendiculars on any tangent to the ellipse x 2 / a 2 + y 2 / b 2 = 1 from two points on the minor axis each at a distance a 2 − b 2 from the centre is

    If the tangent at P ( θ ) on the ellipse 16 x 2 + 11 y 2 = 256 touches the circle x 2 + y 2 + 2 x − 15 = 0 , then θ −

    If the area of the triangle inscribed in the parabola y 2 = 4 a x with one vertex at the vertex of the parabola and other two vertices at the extremities of a focal chord is 5 a 2 / 2 then the length of the focal chord is

    Statement-1: The locus of the point of intersection of the tangents that are at right angles to the hyperbola x 2 36 – y 2 16 = 1 is the circle x 2 + y 2 = 52 Statement-2: Perpendicular tangents to the hyperbola x 2 a 2 – y 2 b 2 = 1 interest on the director circle x 2 + y 2 = a 2 – b 2 a 2 > b 2 of the hyperbola.

    If the tangents at the extremities of a focal chord of the parabola x 2 = 4 a y meet the tangent at the vertex at points whose abcissac are x 1 and x 2 then x 1 x 2 =

    Normal at point (5, 3) to the rectangular hyperbola x y – y – 2 x – 2 = 0 meets the curve at the point whose coordinates are

    A hyperbola H : x 2 9 – y 2 4 = 1 ntersects the circle, C : x 2 + y 2 – 8 x = 0 at the points A and B. Statement-1: 2 x – 5 y + 4 = 0 is a common tangentto both C and H. Statement-2: Circle on AB as a diameter passes through the centre of the hyperbola H.

    If the chords of contacts of the tangents from the points x 1 , y 1 and x 2 , y 2 to the hyperbola 2 x 2 − 3 y 2 = 6 are at right angle, then 4 x 1 x 2 + 9 y 1 y 2 is equal to

    If the normal at the point P ( θ ) to the ellipse x 2 14 + y 2 5 = 1 intersect it again at the point Q ( 2 θ ) , then cos ⁡ θ =

    Let L be a normal to the parabola y 2 = 4 x . If L passes through the point ( 9 , 6 ) , then L is not given by

    Equation of the line joining the foci of the parabolas y 2 = 4 x and x 2 = − 4 y

    The centre C of a variable circle passing through a fixed point ( a , 0 ) , a > 0 , touches the line y = x . Locus of C is a parabola whose directrix is

    If y = mx + 4 is a tangent to both the parabolas y 2 = 4 x and x 2 = 2 b y , then b is equal to

    If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is:

    Let the line y = m x and the ellipse 2 x 2 + y 2 = 1 intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co–ordinate axes at − 1 3 2 , 0 and 0 , β , then β is equal to:

    The locus of a point which divides the line segment joining the point 0 , – 1 and a point on the parabola x 2 = 4 y internally in the ratio 1:2 is:

    Let the line y = m x and the ellipse 2 x 2 + y 2 = 1 intersect at a point P in the first quadrant. the area of the traingle formed by the normal to this ellipse at P with axes is N o r m a l i s p a s sin g t h r o u g h t h e p o i n t – 1 3 2 , 0

    If e 1 and e 2 are the eccentricities of the ellipse, x 2 18 + y 2 4 = 1 and the hyperbola, x 2 9 – y 2 4 = 1 respectively and e 1 , e 2 is a point on the ellipse, 15 x 2 + 3 y 2 = k , then k is equal to:

    Let P i and P i ‘ be the feet of the perpendicular drawn from the foci S and S ‘ on a tangent T i to an ellipse whose length of semi major axis is 20. If ∑ i = 1 10 S P i S ‘ P i 1 = 2560 , then the value of the eccentricity is

    The number of distinct normals that can be drawn from (-2, 1) to the parabola y 2 – 4 x – 2 y – 3 = 0 is

    Let P N be the ordinate of a point P on the hyperbola x 2 ( 97 ) 2 – y 2 ( 79 ) 2 = 1 and the tangent at P meets the transverse axis in T , O is the origin. Then O N . O T 2020 is equal to… (where [.] denotes Greatest integer function)

    If P is a point on the hyperbola x 2 – y 2 = a 2 , C is its centre and S , S ‘ are two foci, then S P ⋅ S ‘ P =

    An ellipse passing through the origin has its foci at (3,4) and (6,8), and length of its semi-minor axis L. The value of L 2 is

    If the equation of the ellipse whose axes are coincident with the coordinate axes and which touches the straight lines 3 x − 2 y − 20 = 0 and x + 6 y − 20 = 0 is x 2 a 2 + y 2 b 2 = 1 , then a+b=

    Let the normal to parabola y 2 = 4 a x at P meets the curve again in Q . If P Q and the normal at Q makes angles α and β respectively with the positive x -axis in positive direction, then tan ⁡ α ( tan ⁡ α + tan ⁡ β ) is equal to

    If P Q is a double ordinate of the hyperbola 16 x 2 − 25 y 2 = 400 be produced on both sides to meet the asymptotes at R and S , then P R ⋅ P S =

    Two straight lines are perpendicular to each other. One of them touches the parabola y 2 = 4 a x + a and the other touches y 2 = 4 b x + b .Their point of intersection lies on the line

    The equation of the parabola having focus at 0 , – 3 and directrix y = 3

    The equation of the parabola having vertex at origin and focus at 3 , 0

    The curve described parametrically by x = t 2 + t + 1 , and y = t 2 − t + 1 represents

    The equation of the parabola whose focus is the point ( 0 , 0 ) and the tangent at the vertex is x − y + 1 = 0 is

    The vertex of the parabola 2 ( x − 1 ) 2 + ( y − 2 ) 2 = ( x + y + 3 ) i s

    In the given figure, a parabola is drawn to pass through the vertices B, C and D of the square are ABCD .If the coordinates of A and C are (2,1) and (2,3), respectively, then focus of this parabola is

    The angle between the tangents drawn from the point ( 1 , 4 ) to the parabola y 2 = 4 x is

    If the normals to the curve y = x 2 at the points P , Q and R nass through the point ( 0 , 3 / 2 ) then the radius of the circle circumscribing Δ P Q R is

    If the length of the latus rectum of the parabola 169 ( x − 1 ) 2 + ( y − 3 ) 2 = ( 5 x − 12 y + 17 ) 2 is L then the value of 13 L is

    y = x + 2 is any tangent to the parabola y 2 = 8 x . The ordinate of the point P on this tangent such that the other tangent from it which is perpendicular to it is

    Consider the locus of centre of circle which touches circle x 2 + y 2 = 4 and line x = 4 . The distance of the vertex of the locus from origin is

    Line y = 2 x − b cuts the parabola y = x 2 − 4 x at points A and B . The value of b for which the ∠ A O B is a right angle is

    If circle ( x − 6 ) 2 + y 2 = r 2 and parabola y 2 = 4 x have maximum number of common chords, then least integral the value of ‘r’ is

    If the vertex of the conic represented by 25 x 2 + y 2 = ( 3 x − 4 y + 12 ) 2 is ( a , b ) then the value of ( b + a ) is

    The locus of the centre of a circle which cuts orthogonally the parabola y 2 = 4 x at ( 1 , 2 ) will pass through

    If y = 2 x − 3 is a tangent to the parabola y 2 = 4 a x − 1 3 , then a is equal to

    From a point ( sin ⁡ θ , cos ⁡ θ ) , if three normals can be drawn to the parabola y 2 = 4 a x , then the value of a is

    Tangents are drawn from any point on the line x + 4 a = 0 to the parabola y 2 = 4 a x . Then the angle subtended by chord of contact at the vertex is

    If the chord of contact of tangents from a point P to the parabola y 2 = 4 a x touches the parabola x 2 = 4 b y , then the locus of P is

    The locus of the point of intersection of two normals to a parabola which are at right angles to one another is

    The vertex of a parabola is the point (a, b) and the latus rectum is of length l . If the axis of the parabola is parallel to the y – axis and the parabola is concave upward, then its equation is

    If the mid point of a chord of the ellipse x 2 16 + y 2 25 = 1 is ( 0 , 3 ) , then length of the chord is

    If the locus of the moving point P ( x , y ) satisfying ( x − 1 ) 2 + y 2 + ( x + 1 ) 2 + ( y − 12 ) 2 = a is ellipse, then the least integral v alue of a is

    The area of the rectangle formed by the perpendiculars from centre of x 2 9 + y 2 4 = 1 to the tangent and normal at the point whose eccentric angle is π 4 is A . The value of 13A is

    A point P lies on the ellipse ( y − 1 ) 2 64 + ( x + 2 ) 2 49 = 1 . If distance of P from one focus is 8 , then its distance from other focus is

    Maximum value of the square of length of chord of the ellipse x 2 8 + y 2 4 = 1 , such that eccentric angles of its extremities differ by π 2 is

    If the distances of one focus of hyperbola from its directrices are 5 and 3, then eccentricity of hyperbola is

    Let E be the ellipse x 2 9 + y 2 4 = 1 and C be the circle x 2 + y 2 = 9 . Let P and Q be the points ( 1 , 2 ) and ( 2 , 1 ) respectively. Then,

    If (5 , 12) and (24, 7) arc the foci of an ellipse passing through the origin, then the eccentricity of the ellipse is

    The equation ( 5 x − 1 ) 2 + ( 5 y − 2 ) 2 = λ 2 − 2 λ + 1 ( 3 x + 4 y − 1 ) 2 represents an ellipse if λ ∈

    The normal at a point P on the ellipse x 2 + 4 y 2 = 16 meets the x -axis at Q . If M is the midpoint of the line segment P Q , then the locus of M intersects the latus rectums of the given ellipse at points

    Consider a branch of the hyperbola x 2 − 2 y 2 − 2 2 x − 4 2 y − 6 = 0 with vertex at the point A . Let B be one of the endpoints of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of triangle ABC is

    Each of the four inequalities given below defines a region in the xy plane. One of these four regions does not have the following property . For any two points x 1 , y 1 and x 2 , y 2 in the region, the point x 1 + x 2 / 2 , y 1 + y 2 / 2 is also in the region. The inequality defining this region is

    At the point of intersection of the rectangular hyperbola x y = c 2 and the parabola y 2 = 4 a x tangents to the rectangular hyperbola and the parabola make angles θ and ϕ , respectively with x -axis, then

    A normal to the hyperbola x 2 4 − y 2 1 = 1 has equal intercepts on the positive x – and y -axes. If this normal touches the ellipse x 2 a 2 + y 2 b 2 = 1 , then a 2 + b 2 is equal to

    The equation of the ellipse with its centre at (1,2), focus at (6,2) and passing through the point (4, 6) is

    The equation of ellipse circumscribing the quadrilateral whose sides are given by x =±2 and y = ±4 and distance between foci is 4 6 , is

    If P = ( x , y ) , F 1 = ( 3 , 0 ) , F 2 = ( − 3 , 0 ) , and 16 x 2 + 25 y 2 = 400 , then P F 1 + P F 2 equals

    The line passing through the extremity A of the major axis and extremi ty B of the minor axis of the ellipse x 2 +9y 2 =9 meets its auxiliary circle at the point M Then the area of the triangle with vertices at A , M , and O ( the origin ) is .

    An ellipse having foci at (3, 3) and (1,4) and passing through the origin has eccentricity equal to

    P and Q are the foci of the ellipse x 2 a 2 + y 2 b 2 = 1 and B is an end of the minor axis. If P B Q is an equilateral triangle, then the eccentricity of the ellipse is

    A man running around a race course notes that the sum of the distances of two flagposts from him is always 10 m and the distance between the flag posts is 8 m . Then the area of the path he encloses in square meters is

    The ratio of the area enclosed by the locus of the midpoint of P S and area of the ellipse x 2 a 2 + y 2 b 2 = 1 is ( P be any point on the ellipse and S be iti focus)

    If x cos ⁡ α + y sin ⁡ α = 4 is tangent to x 2 25 + y 2 9 = 1 , then the value of α is

    The number of values of c such that the straight line y = 4 x + c touches the curve x 2 4 + y 2 1 = 1 is

    The angle between the pair of tangents from the point ( 1 , 2 ) to the ellipse 3 x 2 + 2 y 2 = 5 is

    Length of the perpendicular from the centre of the ellipse 27 x 2 + 9 y 2 = 243 on a tangent drawn to it which makes equal intercepts on the coordinates axes is

    Tangents are drawn from any point on the circle x 2 + y 2 = 41 to the ellipse x 2 25 + y 2 16 = 1 then the angle between the two tangents is

    A common tangent to the conics x 2 = 6 y and 2 x 2 − 4 y 2 = 9 is

    If a tangent of slope 2 of the ellipse x 2 a 2 + y 2 b 2 = 1 is normal to the circle x 2 + y 2 + 4 x + 1 = 0 , then the maximum value of a b is

    If tangent and normal are drawn at the end ‘P’ of a latus rectum in first quadrant of hyperbola x 2 4 − y 2 12 = 1 , meet the transverse axis at T and G respectively, then the area of the triangle P T G is

    If PQ is the focal chord of parabola y 2 = − x and P is ( − 4 , 2 ) , then the slope of tangent at Q is

    A circle passes through the focus of parabola x 2 = 4 y . If the circle touches the parabola at (6,9), then square of its radius is

    The focal chord of the parabola ( y − 2 ) 2 = 16 ( x − 1 ) is a tangent to the circle x 2 + y 2 − 14 x − 4 y + 51 = 0 , then slope of the focal chord can be

    Area of the quadrilateral formed by drawing tangents at the ends of latusrecta of x 2 4 + y 2 1 = 1 is

    Number of integral values of b for which tangent parallel to line y = x + 1 can be drawn to hyperbola x 2 5 − y 2 b 2 = 1 is

    The area of triangle formed by tangents and the chord of contact from ( 3 , 4 ) to y 2 = 2 x is

    x 2 f ( 4 a ) + y 2 f a 2 − 5 = 1 represents an ellipse with major axis as y -axis and ‘ f ‘ is decreasing function, positive for all ‘a’, then a belongs to

    Area of rectangle formed by the tangents drawn to the ellipse x 2 16 + y 2 9 = 1 at ends of its minor axis and tangents drawn to the circle x 2 + y 2 = 16 where it crosses x -axis is

    If circle ( x − 6 ) 2 + y 2 = r 2 and parabola y 2 = 4 x have maximum number of common chords, then least integral value of r is

    For a point P on the ellipse 9 x 2 + 36 y 2 = 324 , with foci S and S ′ , value of SP + S ′ P =

    The equation of the common tangent to the curves y 2 = 8 x and x y = − 1 , is

    If the eccentricity of the hyperbola x 2 − y 2 sec 2 ⁡ θ = 5 is 3 times the eccentricity of the ellipse x 2 sec 2 ⁡ θ + y 2 = 25 , and smallest positive value of θ is π P then twice the value of ‘p’is

    If e 1 , and e 2 are respectively the eccentricities of the conics x 2 25 − y 2 11 = 1 and x 2 16 + y 2 7 = 1 then e 1 e 2 is equal to

    Let L be a normal to the parabola y 2 = 4x. If L passes through the point (9, 6), then L is not given by

    α , β are the eccentric angles of the extremities of a focal chord of the ellipse x 2 / 16 + y 2 / 9 = 1 then tan ⁡ ( α / 2 ) tan ⁡ ( β / 2 ) =

    Let P be a point on the hyperbola x 2 81 − y 2 49 = 1 .The tangent at P meets the transverse axis at T, N is the foot of the perpendicular from P to the transverse axis. If O is the origin, then ON.OT is equal to.

    The length of the latus rectum of the parabola 289 ( x − 3 ) 2 + ( y − 1 ) 2 = ( 15 x − 8 y + 13 ) 2 is equal to

    If t 1 + t 2 + t 3 = − t 1 t 2 t 3 then the orthocentre of the triangle formed by the points A at 1 , a t 1 + t 2 B at 2 t 3 , a t 2 + t 3 , C at 3 t 1 , a t 3 + t 1 , lies on

    The equation of the line touching both the parabola y 2 = 4x and x 2 = − 32 y is ax + by + c = 0 . Then the value of a + b +c is

    If on a given base BC [B(0, 0) and C(2,0)], a triangle is described such that the sum of the tangents of the base angles is 4, then the equation of the locus of the opposite vertex A is a parabola whose directrix is, y = k. The value of k is

    The value of a for the ellipse x 2 a 2 + y 2 b 2 = 1 , ( a > b ) if the extremities of the latusrectum of the ellipse having positive ordinates lie on the parabola x 2 = − 2 ( y − 2 ) is

    If tangents drawn from the point (a, 2) to the hyperbola x 2 16 − y 2 9 = 1 are perpendicular, then the value of a 2 is

    The length of focal chord to the parabola y 2 = 12 x drawn from the point (3, 6) on it is .

    An ellipse passing through the origin has its foci (3, 4) and (6, 8). The length of its semi-minor axis is 6. Then the value of b / 2 is .

    If the normal at P on the hyperbola x 2 a 2 − y 2 b 2 = 1 meets the transverse axis at G , S is a foci and e the eccentricity of the hyperbola then S G : S P is equal to

    P is a point on the hyperbola x 2 a 2 − y 2 b 2 = 1 S S and S ′ are its foci. Statement-1: Product of the lengths of the perpendiculars from S and S ′ on the tangent at P is equal to Statement-2: P S − P S ′ = 2 a .

    Let y 2 = 16 x be a given parabola and L be an extremity of its latus rectum in the first quadrant. If a chord is drawn through L with slope-1, then the length of this chord is:

    The locus of the middle points of the normal chords of the rectangular hyperbola x 2 − y 2 = a 2 is

    Consider a branch of the hyperbola x 2 − 2 y 2 − 2 2 x − 4 2 y − 6 = 0 with vertex at the point A . let B be one of the end points of its latus rectum. If C is the focus of the hyperbola near A , then area of the ∆ A B C is:

    Statement-1: If the foci of a hyperbola are at the points (4, 1) and (–6, 1), eccentricity is 5/4 then the length of the transverse axis is 8. Statement-2: Distance between the foci of a hyper bola is equal to the product of its eccentricity and the length of the transverse axis

    If the distance between two directrices of a rectangular hyperbola is 15, then the distance between its foci in units is:

    The coordinates of a point common to a directrix and an asymptote of the hyperbola x 2 25 − y 2 16 = 1 are

    Let ( x , y ) be any point on the parabola y 2 = 4 x . Let P be the point that divides the line segment from ( 0 , 0 ) to ( x , y ) in the ratio 2 : 3 . Then locus of P is

    Let P ( x , y ) be a variable point such that ( x − 3 ) 2 + ( y − 2 ) 2 − ( x − 6 ) 2 + ( y + 2 ) 2 = 3 . Statement-1: P traces a hyperbola whose eccentricity is 5 3 . Statement-2: P traces a hyperbola such that the equation of its conjugate axis is 6 x – 8 y = 27

    The product of the perpendiculars from the foci on any tangent to the hyperbola x 2 64 − y 2 9 = 1 is

    Statement-1: Two tangents drawn from any point on he hyperbola x 2 − y 2 = a 2 − b 2 to the ellipse x 2 a 2 + y 2 b 2 =1 make complementary angles with the axis of the ellipse. Statement-2: If two lines make complementary angles with the axis of x then the product of their slopes is 1.

    If the normals at P , Q , R on the rectangular hyperbola x y = c 2 intersect at a point S on the hyperbola, then centroid of the triangle P Q R is at

    The curve represented by x = 5 t + 1 t , y = t − 1 t , t ≠ 0 is

    The point of intersection of the normals to the parabola y 2 = 4 x at the ends of its latus rectum is

    The locus of the mid-points of the chords the parabola x 2 = 4 p y having slope m is a:

    Statement-1: If the angle between two asymptotes of a hyperbola x 2 a 2 − y 2 b 2 = 1 is π 3 its eccentricity is 2 3 . Statement 2: Angles between the asymptotes of the hyperbola x 2 a 2 − y 2 b 2 = 1 are 2 tan − 1 ⁡ b a or π − 2 tan − 1 ⁡ b a .

    If a diameter of a hyperbola meets the hyperbola in real points then

    chord is drawn through the focus of the parabola y 2 = 6 x such that its distance from the vertex of this parabola is 5 2 then its slope can be:

    An equation of the parabola whose focus is (–3, 0) and the directrix x + 5 = 0 is:

    The point of intersection of the normals to the parabola y 2 = 4 x at the ends of its latusrectum is

    Let Q be the foot of the perpendicular from the origin O to the tangent at a point p ( α , β ) on the parabola y 2 = 4 a x , and S be the focus of the parabola, then ( O Q ) 2 ( S P ) is equal to

    If P is the point ( 1 , 0 ) and Q lies on the parabola y 2 = 36 x then the locus of the mid point of P Q is

    If a tangent to the parabola y 2 = 4 x makes an angle π / 4 with the positive direction of the axis of x , then the coordinates of the point of contact with the parabola are:

    The tangent at a point P on the parabola y 2 = 8 x meets the directrix of the parabola at Q such that distance of Q from the axis of the parabola is 3. Then the coordinates of P cannot be

    Length of the common chord of the parabola y 2 = 8 x and the circle x 2 + y 2 − 2 x − 4 y = 0 is

    The slope of the line touching both the parabolas y 2 = 4 x and x 2 = − 32 y is

    An equation of the latus rectum of the parabola x 2 + 4 x + 2 y = 0 is

    Distance of a point P on the parabola y 2 = 48 x from the focus is l and its distance from the tangent at the vertex is d , then l – d is equal to

    Locus of the mid points of the chords of the parabola y 2 = 8 x which touch the circle x 2 + y 2 = 4 is

    If θ is the angle between the tangents to the parabola y 2 = 12 x passing through the point (–1, 2) then | tan ⁡ θ | is equal to

    Let L 1 be the length of the common chord of the curves x 2 + y 2 = 9 and y 2 = 8 x , and L 2 be the length of the latus rectum of y 2 = 8 x , then

    If m is the slope of a common tangent of the parabola y 2 = 16 x and the circle x 2 + y 2 = 8 , then m 2 is equal to

    P 1 : y 2 = 49 x and P 2 : x 2 = 4 a y are two parabolas. Equation of a tangent to the parabola P 1 at a point where it intersects the parabola P 2 is:

    Equation of a normal to the parabola y 2 = 32 x passing through its focus is

    The length of the chord of the parabola y 2 = 4 a x whose equation is. y − x 2 + 4 a 2 = 0 is

    If the normal drawn from the point on the axis of the parabola y 2 = 8 a x whose distance from the focus is 8 a , and which is not parallel to either axes, makes an angle θ with the axis of x , then θ is equal to

    A point on the parabola y 2 = 18 x at which the ordinate increases at twice the rate of the abscissa is

    If P be the point ( 1 , 0 ) and Q , a point on the locus y 2 = 8 x . The locus of the mid point of P Q is:

    For the parabola y 2 + 8 x − 12 y + 20 = 0 which of the following in not correct

    The coordinates of the end point of the latus rectum of the parabola ( y − 1 ) 2 = 2 ( x + 2 ) which does not lie on the line 2 x + y + 3 = 0 are

    If P is the length of the perpendicular from a focus upon the tangent at any point P of the ellipse x 2 / a 2 + y 2 / b 2 = 1 and r r is the distance of P from the focus, then 2 a r − b 2 p 2 =

    The point of contact of the tangent to the parabola y 2 = 9 x which passes through the point (4, 10) and makes an angle θ with the axis of the parabola such that tan ⁡ θ > 2 is

    The focus of the parabola 4 y 2 + 12 x − 20 y + 67 = 0

    If y = x and 3 y + 2 x = 0 are the equations of a pair of conjugate diameters of an ellipse, then the eccentricity of the ellipse is

    If a, b are the eccentric angles of the extremities of a focal chord of the ellipse x 2 / 16 + y 2 / 9 = 1 then tan ⁡ ( α / 2 ) tan ⁡ ( β / 2 ) =

    Equation of the normal at a point on the parabola y 2 = 36 x , whose ordinate is three times its abcissac is-

    A chord is drawn through the focus of the parabola y 2 = 6 x such than its distance from the vertex of this parabola is 5 2 , then its slope can be

    Which of the following parametric equation does not represent a parabola

    If the line k x + y = 4 touches the parabola y = x − x 2 then the point of contact is

    An equation of a common tangent to the parabola y 2 = 4 x and x 2 = 4 y is

    If the normal at P to the parabola y 2 = 4 a x meets the parabola again at Q such that P Q subtends a right angle at the vertex of the parabola, then the coordinates of P are

    The locus of the points of intersection of the tangents at the extremities of the chords of the ellipse x 2 + 2 y 2 = 6 which touch the ellipse x 2 + 4 y 2 = 4 is

    The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y 2 = 4 a x is another parabola with directrix

    If P , Q , R are three points on a parabola y 2 = 4 a x whose ordinates are in geometrical progression, then the tangents at P and R meet on

    Equation of a common tangent to the curves y 2 = 8 x and x y = – 1 is

    If perpendiculars are drawn on any tangent to a parabola y 2 = 4 a x from the points ( a ± k , 0 ) on the axis. The difference of the squares of their lengths is

    Length of the tangent drawn from an end of the latus rectum of the parabola y 2 = 4 a x to the circle of radius a touching externally the parabola at the vertex is

    If chords of contact of the tangent from two points x 1 , y 1 and x 2 , y 2 to the ellipse x 2 a 2 + y 2 b 2 = 1 are at right angles, then x 1 x 2 a 2 × y 1 y 2 b 2 is equal to

    Equation of the tangent at a point P on the parabola y 2 = 4 a x the normal at which is at a distance a 5 / 4 from the focus of the parabola is

    y = m x + c c is a normal to the ellipse x 2 / a 2 + y 2 / b 2 = 1 if c 2 is equal to

    An isosceles triangle is inscribed in the parabola y 2 = 4 a x with its base as the line joining the vertex of the parabola and positive end of the latus rectum of the parabola. If a t 2 , 2 a t is the vertex of the triangle then

    Equation of a family of circles passing through the extremities of the latus rectum of the parabola y 2 = 4 a x , g being a parametric, is

    A tangent at any point to the ellipse 4 x 2 + 9 y 2 = 36 is cut by the tangent at the extremities of the major axis at T ′ T and ‘. The circle on T T ′ as diameter passes through the point.

    P is a point on the parabola y 2 = 4 a x whose ordinate is equal to its abscissa and P Q is a focal chord, R and S are the feet of the perpendiculars from P and Q respectively on the tangent at the vertex, T is the foot of the perpendicular from Q to P R , area of the triangle P T Q is

    Tangents at the extremities of a focal chord of a parabola intersect

    P is a point on the locus of the mid-points of the chords of the parabola y 2 = 4 a x passing through the vertex of the parabola; S is the focus of the parabola and the line joining S and P is produced to meet the parabola at Q . If the ordinate of P is equal to its abscissa, coordinates of Q are

    Locus of the point of intersection of the normals to the parabola y 2 = 16 x which are at right angles is

    In an ellipse, the distance between the foci is 6 and length of semi minor axis is 4 , then eccentricity of the ellipse is

    The length of the perpendiculars from the focus and the extremities of a focal chord of a parabola on the tangent at the vertex form

    Equation of an ellipse with centre at the origin passing through ( 5 , 0 ) and having eccentricity 2 / 3 is.

    If the eccentric angles of two points P and Q on the ellipse x 2 a 2 + y 2 b 2 are α , β such that α + β = π 2 then,the locus of the point of intersection of the normals at P and Q is

    If the latus rectum of a hyperbola subtend an angle of 60° at the other focus, then eccentricity of the hyperbola is

    The tangent to the ellipse 3 x 2 + 16 y 2 = 12 , at the point (1, 3/4), intersects the curve y 2 + x = 0 at :

    Let L be a normal to the parabola y 2 = 4 x . If L passes through the point (9, 6), then equation of L can not be

    Equation of a tangent to the ellipse x 2 36 + y 2 25 = 1 passing through the point where a directrix of the ellipse meets the positive x-axis is

    E : x 2 a 2 + y 2 b 2 = 1 and P : y 2 = 4 b x , a > b . Statement 1: The tangent at the positive end of the minor axis of the ellipse. E passes through the positive end of the latus rectum of the parabola P . Statement 2: If the latus rectum of the parabola P is same as that of the ellipse E , then eccentricity of E is 1 / 2

    Statement 1: A normal to the hyperbola with eccentricity 3, meets the transverse axis and conjugate axis at P and Q respectively. The locus of the mid-point of P Q is a hyperbola with eccentricity 3 2 2 Statement 2: Eccentricity of the hyperbol 8 x 2 − y 2 = 8 a 2 is 3 2 2 .

    Statement 1: If the extremities of the latus rectum of the ellipse x 2 a 2 + y 2 b 2 = 1 (a> b), having positive ordinates lies on the parabola x 2 = − 2 ( y − 2 ) , then, a = 2 Statement 2: If the length of the latus rectum of the ellipse x 2 a 2 + y 2 b 2 = 1 is equal to the distance between the foci, then the eccentricity e of the ellipse satisfies e 2 + e − 1 = 0

    Statement-1: Equation of a circle on the ends of a latus rectum of the hyperbola x 2 16 − y 2 9 = 1 as a diameter is 16 x 2 + 16 y 2 ± 160 x + 319 = 0 Statement-2: Focus of the parabola y 2 = 20 x coin cides with a focus of the hyperbola x 2 16 − y 2 9 = 1

    An ellipse has eccentricity 1/2 and a focus at the point P(1/2, 1). One of its directrix is the common tangent near to the point P, to the circle x 2 + y 2 = 1 and the hyperbola x 2 − y 2 = 1 the equation of the ellipse is

    The asymptotes of x y = h x + ky are

    H 1 : x y = c 2 and H 2 : x y = k 2 are two different hyperbolas. From a point on H 1 , tangents are drawn to H 2 .Area of the triangle formed by the chord of contact and the asymptote to H 2 is

    Locus of the mid-point of the chord of the hyperbola x 2 a 2 – y 2 b 2 = 1 which is a tangent to the circle x 2 + y 2 = c 2 is

    e 1 , e 2 are respectively the eccentricities of the hyperbola x 2 – y 2 cosec 2 θ = 5 and the ellipse x 2 cosec 2 θ + y 2 = 5 . If 0 < θ < π / 2 and e 1 = 7 e 2 , then θ is equal to

    A and B are two points on the hyperbola x 2 a 2 − y 2 b 2 = 1 O is the centre. If O A is perpendicular to O B then 1 ( O A ) 2 + 1 ( O B ) 2 is equal to

    If θ is an angle between the two asymptotes of the hyperbola x 2 a 2 − y 2 b 2 = 1 , then cos ⁡ θ 2 is equal to

    Let E be the ellipse x 2 9 + y 2 4 = 1 and C be the circle x 2 + y 2 = 9 . Let P ( 1 , 2 ) and Q ( 2 , 1 ) be two points, then

    If l x + m y + n = 0 is an equation of the line joining the extremities of a pair of semi-conjugate diameters of t the ellipse x 2 9 + y 2 4 = 1 , then 9 l 2 + 4 m 2 n 2 is equal to

    Let P be a point in the first quadrant lying on the ellipse 9 x 2 + 16 y 2 = 144 , such that the tangent at P to the ellipse is inclined at an angle 135° to the positive direction of x – axis. Then the coordinates of P are:

    P N is an ordinate of the parabola y 2 = 9 x . A straight line is drawn through the mid-point M of P N parallel to the axis of the parabola meeting the parabola at Q . N Q meets the tangent at the vertex A , at a point T , then A T / N P =

    A parabola has the origin as its focus and the line x = 2 as the directrix. Then the vertex of the parabola is at

    y = − 2 x + 12 a is a normal to the parabola y 2 = 4 a x at the point whose distance from the directrix of the parabola is

    y 2 = 16 x is a parabola and x 2 + y 2 = 16 is a circle . Then

    A line bisecting the ordinate P N of a point P ( a t 2 , 2 a t ) , t > 0 , on the parabola y 2 = 4 a x is drawn parallel to the axis to meet the curve at Q . If N Q meets the tangent at the vertex at the point T, then the coordinates of T are.

    A line bisecting the ordinate P N of a point P ( a t 2 , 2 a t ) , t > 0 , on the parabola y 2 = 4 a x is drawn parallel to the axis to meet the curve at Q . If N Q meets the tangent at the vertex at the point T, then the coordinates of T are.

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