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Conic Sections · Question 30

JEE Main 2026 (Online) 28th January Evening ShiftMCQMedium📈 Topic in 39% papers+4−1
1

Let the circle x2 + y2 = 4 intersect x-axis at the points A(a, 0), a > 0 and B(b, 0). Let P(2 , 2 ), 0 < < 2 and Q(2 , 2 ) be two points such that ( - ) = 2. Then the point of intersection of AQ and BP lies on :

  1. x2 + y2 - 4x - 4 = 0
  2. x2 + y2 - 4x - 4y = 0
  3. x2 + y2 - 4x - 4y - 4 = 0
  4. x2 + y2 - 4y - 4 = 0

Answer

(4) x2 + y2 - 4y - 4 = 0

Solution

A(2, 0), B(−2, 0), so AB is a diameter of the circle (centre O).

- = 2 POQ = 90

PAQ = 12 POQ = 45 (angle at the circumference)

AB is a diameter APB = 90

Let R be the point of intersection of AQ and BP. In APR: APR = 90 , PAR = 45

ARB = 45 (constant)

R lies on a circle through A and B on which chord AB subtends 45 , i.e. 90 at its centre.

Centre (0, k) on the perpendicular bisector of AB, radius = AB 2 = 2 2

k2 + 4 = 8 k = 2 (R is above the x-axis since P is in the first quadrant)

locus: x2 + (y - 2)2 = 8 x2 + y2 - 4y - 4 = 0

Check: = 3 gives R(1 + 3, 1 + 3), which satisfies it.

Answer: (4)

JEE Main 2026 (Online) 28th January Evening Shift

Keep practising — Conic Sections

2

An ellipse has its center at (1, -2), one focus at (3, -2) and one vertex at (5, -2). Then the length of its latus rectum is :

  1. 6
  2. 6 3
  3. 16 3
  4. 4 3
JEE Main 2026 (Online) 28th January Evening Shift
3

Let A be the focus of the parabola y2 = 8x. Let the line y = mx + c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is ( 73, 43 ), then (BC)2 is equal to :

  1. 89
  2. 80
  3. 32
  4. 41
JEE Main 2026 (Online) 28th January Evening Shift
4

Let the ellipse E : x2144 + y2169 = 1 and the hyperbola H : x216 - y2 2 = -1 have the same foci. If e and L respectively denote the eccentricity and the length of the latus rectum of H, then the value of 24(e + L) is :

  1. 296
  2. 126
  3. 67
  4. 148
JEE Main 2026 (Online) 28th January Evening Shift
5

Let a circle of radius 4 pass through the origin O, the points A(- 3a, 0) and B(0, - 2b), where a and b are real parameters and ab 0. Then the locus of the centroid of OAB is a circle of radius

  1. 73
  2. 113
  3. 53
  4. 83
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6

Let each of the two ellipses E1: x2a2 + y2b2 = 1, (a > b) and E2: x2A2 + y2B2 = 1, (A < B) have eccentricity 45. Let the lengths of the latus recta of E1 and E2 be l1 and l2, respectively, such that 2l12 = 9l2. If the distance between the foci of E1 is 8, then the distance between the foci of E2 is

  1. 965
  2. 85
  3. 165
  4. 325
JEE Main 2026 (Online) 24th January Morning Shift
7

Let the image of parabola x2 = 4y, in the line x - y = 1 be (y + a)2 = b(x - c), a, b, c N. Then a + b + c is equal to

  1. 12
  2. 8
  3. 6
  4. 4
JEE Main 2026 (Online) 24th January Evening Shift
8

Let the length of the latus rectum of an ellipse x2a2 + y2b2 = 1, (a > b), be 30. If its eccentricity is the maximum value of the function f(t) = - 34 + 2t - t2, then (a2 + b2) is equal to

  1. 276
  2. 516
  3. 256
  4. 496
JEE Main 2026 (Online) 24th January Evening Shift
9

Let (h, k) lie on the circle C : x2 + y2 = 4 and the point (2h + 1, 3k + 2) lie on an ellipse with eccentricity e. Then the value of 5e2 is equal to .

JEE Main 2026 (Online) 24th January Evening Shift
10

Let the domain of the function f(x) = 3 5 7 (9x - x2 - 13 ) be the interval (m, n). Let the hyperbola x2a2 - y2b2 = 1 have eccentricity n3 and the length of the latus rectum 8m3. Then b2 - a2 is equal to :

  1. 7
  2. 9
  3. 11
  4. 5
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11

Let the line y - x = 1 intersect the ellipse x22 + y21 = 1 at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is :

  1. - -1 ( 14 )
  2. 2 + -1 ( 14 )
  3. 2 + 2 -1 ( 14 )
  4. 2 - -1 ( 14 )
JEE Main 2026 (Online) 23rd January Morning Shift

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