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JEE Main Conic Sections questions

66 Mathematics questions on Conic Sections as asked in JEE Main, newest first. 49 come with a worked solution.

  1. The equations of the sides AB and AC of a triangle ABC are (λ + 1)x + λ y = 4 and λ x + (1 - λ)y + λ = 0 respectively. Its vertex A is on the y-axis and its orthocentre is (1,2). The length of the tangent from the point…

    2023

  2. The equations of two sides of a variable triangle are x = 0 and y = 3, and its third side is a tangent to the parabola y^2 = 6x. The locus of its circumcentre is:

    2023

  3. Let a tangent to the curve y^2 = 24x meet the curve xy = 2 at the points A and B. Then the mid points of such line segments AB lie on a parabola with the

    2023 · with solution

  4. The vertices of a hyperbola H are (± 6, 0) and its eccentricity is dfrac√(5)2. Let N be the normal to H at a point in the first quadrant and parallel to the line √(2)x + y = 2√(2). If d is the length of the line segment…

    2023

  5. Let P(4, -4) and Q(9, 6) be two points on the parabola, y^2 = 4x and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of triangle PXQ is maximum. Then, this…

    2019

  6. If the parabolas y^2 = 4b(x - c) and y^2 = 8ax have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c)?

    2019

  7. If the area of the triangle whose one vertex is at the vertex of the parabola, y^2 + 4(x - a^2) = 0 and the other two vertices are the points of intersection of the parabola and Y-axis, is 250 sq units, then a value of…

    2019

  8. If the line ax + y = c, touches both the curves x^2 + y^2 = 1 and y^2 = 4√(2)x, then |c| is equal to

    2019 · with solution

  9. An ellipse, with foci at (0, 2) and (0, -2) and minor axis of length 4, passes through which of the following points?

    2019 · with solution

  10. If the normal to the ellipse 3x^2 + 4y^2 = 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent to the ellipse at P passes through Q(4, 1) then PQ is equal to

    2019

  11. Let P be the point of intersection of the common tangents to the parabola y^2 = 12x and the hyperbola 8x^2 - y^2 = 8. If S and S' denotes the foci of the hyperbola where S lies on the positive X-axis then P divides SS'…

    2019 · with solution

  12. Let O(0, 0) and A(0, 1) be two fixed points, then the locus of a point P such that the perimeter of triangle AOP is 4, is

    2019 · with solution

  13. The tangents to the curve y = (x-2)^2 - 1 at its points of intersection with the line x - y = 3, intersect at the point

    2019 · with solution

  14. In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at (0, 5√(3)), then the length of its latus rectum is

    2019 · with solution

  15. The tangent and normal to the ellipse 3x^2 + 5y^2 = 32 at the point P(2, 2) meets the X-axis at Q and R, respectively. Then, the area (in sq units) of the triangle PQR is

    2019 · with solution

  16. If the line y = mx + 7√(3) is normal to the hyperbola (x^2)/24 - (y^2)/18 = 1, then a value of m is

    2019

  17. If 5x + 9 = 0 is the directrix of the hyperbola 16x^2 - 9y^2 = 144, then its corresponding focus is

    2019

  18. The area (in sq units) of the smaller of the two circles that touch the parabola, y^2 = 4x at the point (1, 2) and the X-axis is

    2019 · with solution

  19. If the line x - 2y = 12 is tangent to the ellipse (x^2)/(a^2) + (y^2)/(b^2) = 1 at the point (3, (-9)/2 ), then the length of the latus rectum of the ellipse is

    2019 · with solution

  20. Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If triangle S'BS is a right angled triangle with right angle at B and area (triangle S'BS) = 8 sq units, then the length of a…

    2019

  21. If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is

    2019 · with solution

  22. If a directrix of a hyperbola centred at the origin and passing through the point (4, -2√(3)) is 5x = 4√(5) and its eccentricity is e, then

    2019 · with solution

  23. The equation of a tangent to the parabola, x^2 = 8y, which makes an angle θ with the positive direction of X-axis, is

    2019 · with solution

  24. Let the length of the latus rectum of an ellipse with its major axis along X-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one…

    2019 · with solution

  25. If the tangent to the parabola y^2 = x at a point (α, β), (β > 0) is also a tangent to the ellipse x^2 + 2y^2 = 1, then α is equal to

    2019

  26. If the vertices of a hyperbola be at (-2, 0) and (2, 0) and one of its foci be at (-3, 0), then which one of the following points does not lie on this hyperbola?

    2019

  27. The equation of a tangent to the hyperbola 4x^2 - 5y^2 = 20 parallel to the line x - y = 2 is

    2019 · with solution

  28. Equation of a common tangent to the circle, x^2 + y^2 - 6x = 0 and the parabola, y^2 = 4x, is

    2019 · with solution

  29. If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is

    2019 · with solution

  30. Equation of a common tangent to the parabola y^2 = 4x and the hyperbola xy = 2 is

    2019

  31. Let 0 < θ < (π)/2. If the eccentricity of the hyperbola (x^2)/(cos^2θ) - (y^2)/(sin^2θ) = 1 is greater than 2, then the length of its latus rectum lies in the interval

    2019 · with solution

  32. Tangents are drawn to the hyperbola 4x^2 - y^2 = 36 at the points P and Q. If these tangents intersect at the point T(0, 3), then the area (in sq units) of triangle PTQ is

    2018 · with solution

  33. Tangent and normal are drawn at P(16, 16) on the parabola y^2 = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and ∠ CPB = θ, then…

    2018 · with solution

  34. If a hyperbola passes through the point P(√(2), √(3)) and has foci at (± 2, 0), then the tangent to this hyperbola at P also passes through the point

    2017 · with solution

  35. The eccentricity of an ellipse whose centre is at the origin is 1/2. If one of its directrices is x = -4, then the equation of the normal to it at (1, 3/2 ) is

    2017 · with solution

  36. The eccentricity of the hyperbola whose length of the latusrectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is

    2017 · with solution

  37. Let P be the point on the parabola, y^2 = 8x, which is at a minimum distance from the centre C of the circle, x^2 + (y+6)^2 = 1. Then, the equation of the circle, passing through C and having its centre at P is

    2016 · with solution

  38. Let O be the vertex and Q be any point on the parabola x^2 = 8y. If the point P divides the line segment OQ internally in the ratio 1:3, then the locus of P is

    2015 · with solution

  39. The area (in sq units) of the quadrilateral formed by the tangents at the end points of the latusrectum to the ellipse (x^2)/9 + (y^2)/5 = 1 is

    2015 · with solution

  40. The slope of the line touching both the parabolas y^2 = 4x and x^2 = -32y is

    2014 · with solution

  41. The locus of the foot of perpendicular drawn from the centre of the ellipse x^2 + 3y^2 = 6 on any tangent to it is

    2014 · with solution

  42. The equation of the circle passing through the foci of the ellipse (x^2)/16 + (y^2)/9 = 1 and having centre at (0, 3) is

    2013 · with solution

  43. The ellipse E_1: (x^2)/9 + (y^2)/4 = 1 is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E_2 passing through the point (0, 4) circumscribes the rectangle R. The eccentricity…

    2012 · with solution

  44. Let (x, y) be any point on the parabola y^2 = 4x. Let P be the point that divides the line segment from (0, 0) to (x, y) in the ratio 1:3. Then, the locus of P is

    2011 · with solution

  45. Let P(6, 3) be a point on the hyperbola (x^2)/(a^2) - (y^2)/(b^2) = 1. If the normal at the point P intersects the X-axis at (9, 0), then the eccentricity of the hyperbola is

    2011 · with solution

  46. The normal at a point P on the ellipse x^2 + 4y^2 = 16 meets the X-axis at Q. If M is the mid-point of the line segment PQ, then the locus of M intersects the latusrectum of the given ellipse at the points

    2009 · with solution

  47. The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse x^2 + 9y^2 = 9 meets its auxiliary circle at the point M. Then, the area (insqunits) of the triangle with…

    2009

  48. Consider a branch of the hyperbola x^2 - 2y^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 latusrectum. If C is the focus of the hyperbola nearest to the point A, then the…

    2008

  49. A hyperbola, having the transverse axis of length 2sinθ, is confocal with the ellipse 3x^2 + 4y^2 = 12. Then, its equation is

    2007

  50. Axis of a parabola is y = x and vertex and focus are at a distance √(2) and 2√(2) respectively from the origin. Then, equation of the parabola is

    2006

  51. If e_1 is the eccentricity of the ellipse (x^2)/16 + (y^2)/25 = 1 and e_2 is the eccentricity of the hyperbola passing through the foci of the ellipse and e_1 e_2 = 1, then equation of the hyperbola is

    2006 · with solution

  52. If the line 2x + √(6)y = 2 touches the hyperbola x^2 - 2y^2 = 4, then the point of contact is

    2004 · with solution

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

    2004 · with solution

  54. Tangents are drawn to the ellipse x^2 + 2y^2 = 2, then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is

    2004 · with solution

  55. Tangent is drawn to ellipse (x^2)/27 + y^2 = 1 at (3√(3)cosθ, sinθ) (where, θ ∈ (0, π/2)). Then, the value of θ such that the sum of intercepts on axes made by this tangent is minimum, is

    2003 · with solution

  56. The equation of the common tangent to the curves y^2 = 8x and xy = -1 is

    2002 · with solution

  57. If a > 2b > 0, then positive value of m for which y = mx - b√(1 + m^2) is a common tangent to x^2 + y^2 = b^2 and (x - a)^2 + y^2 = b^2 is

    2002 · with solution

  58. The equation of the directrix of the parabola y^2 + 4y + 4x + 2 = 0 is

    2001 · with solution

  59. If x + y = k is normal to y^2 = 12x, then k is

    2000 · with solution

  60. If the line x - 1 = 0 is the directrix of the parabola y^2 - kx + 8 = 0, then one of the values of k is

    2000 · with solution

  61. If x = 9 is the chord of contact of the hyperbola x^2 - y^2 = 9, then the equation of the corresponding pair of tangents is

    1999 · with solution

  62. The curve described parametrically by x = t^2 + t + 1, y = t^2 - t + 1 represents

    1999 · with solution

  63. If P = (x, y), F_1 = (3, 0), F_2 = (-3, 0) and 16x^2 + 25y^2 = 400, then PF_1 + PF_2 equals

    1998 · with solution

  64. The number of values of c such that the straight line y = 4x + c touches the curve (x^2)/4 + y^2 = 1 is

    1998 · with solution

  65. The equation (x^2)/(1-r) - (y^2)/(1+r) = 1, |r| < 1 represents

    1981 · with solution

  66. The tangent to the parabola y^2 = 4x at the point where it intersects the circle x^2 + y^2 = 5 in the first quadrant, passes through the point

    with solution

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