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Conic Sections · Question 84

JEE Main 2025 (Online) 29th January Morning ShiftMCQMedium📈 Topic in 61% papers+4−1
1

Let the ellipse E1 : x2a2 + y2b2 = 1, a > b and E2 : x2A2 + y2B2 = 1, A < B have same eccentricity 1 3. Let the product of their lengths of latus rectums be 32 3 and the distance between the foci of E1 be 4. If E1 and E2 meet at A, B, C and D, then the area of the quadrilateral ABCD equals :

  1. 24 65
  2. 18 65
  3. 6 6
  4. 12 65

Answer

(1) 24 65

Solution

For E1: 2ae = 4 a = 2 3, a2 = 12

b2 = a2(1 - e2) = 12 23 = 8

Latus rectum of E1 = 2b2a = 162 3 = 8 3

Latus rectum of E2 = 32 3 8 3 = 4

For E2 (major axis along y-axis): A2 = B2(1 - e2) = 23B2

2A2B = 4 4B3 = 4 B2 = 9, A2 = 6

Intersection: x212 + y28 = 1 and x26 + y29 = 1

2x2 + 3y2 = 24 and 3x2 + 2y2 = 18

x2 = 65, y2 = 365

ABCD is a rectangle with sides 2|x| and 2|y|.

Area = 4|x||y| = 4 65 365 = 24 65

Answer: (1)

JEE Main 2025 (Online) 29th January Morning Shift

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2

Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at the points A and B, then (AB)2 is equal to :

  1. 384
  2. 392
  3. 96
  4. 192
JEE Main 2025 (Online) 29th January Morning Shift
3

Let the line x + y = 1 meet the circle x2 + y2 = 4 at the points A and B. If the line perpendicular to AB and passing through the mid-point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to :

  1. 14
  2. 3 7
  3. 2 14
  4. 5 7
JEE Main 2025 (Online) 29th January Morning Shift
4

Let ABCD be a trapezium whose vertices lie on the parabola y2 = 4x. Let the sides AD and BC of the trapezium be parallel to y-axis. If the diagonal AC is of length 254 and it passes through the point (1, 0), then the area of ABCD is

  1. 758
  2. 1258
  3. 252
  4. 754
JEE Main 2025 (Online) 28th January Morning Shift
5

Let the equation of the circle, which touches x-axis at the point (a, 0), a > 0 and cuts off an intercept of length b on y-axis be x2 + y2 - x + y + = 0. If the circle lies below x-axis, then the ordered pair (2a, b2) is equal to

  1. ( , 2 + 4 )
  2. ( , 2 - 4 )
  3. ( , 2 - 4 )
  4. ( , 2 + 4 )
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6

Let E1 : x29 + y24 = 1 be an ellipse. Ellipses Ei's are constructed such that their centres and eccentricities are same as that of E1, and the length of minor axis of Ei is the length of major axis of Ei+1 (i 1). If Ai is the area of the ellipse Ei, then 5 ( i=1 Ai ), is equal to .

JEE Main 2025 (Online) 28th January Morning Shift
7

If the midpoint of a chord of the ellipse x29 + y24 = 1 is ( 2, 4/3), and the length of the chord is 2 3, then is:

  1. 26
  2. 18
  3. 22
  4. 20
JEE Main 2025 (Online) 28th January Evening Shift
8

If A and B are the points of intersection of the circle x2 + y2 - 8x = 0 and the hyperbola x29 - y24 = 1 and a point P moves on the line 2x - 3y + 4 = 0, then the centroid of PAB lies on the line:

  1. x + 9y = 36
  2. 9x - 9y = 32
  3. 4x - 9y = 12
  4. 6x - 9y = 20
JEE Main 2025 (Online) 28th January Evening Shift
9

Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If d denotes the distance between A and B, and a denotes the area of SAB, where S is the focus of the parabola y2 = 4x, then the value of (a + d) is .

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