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Conic Sections

JEE Main 2025 (Online) 29th January Morning ShiftMCQMedium📈 Topic in 47% papers+4−1
Ques. 1JEE Main 2025, 29 January (Morning Shift)

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 gives a = 2 3, a2 = 12, and b2 = a2(1 - e2) = 12 23 = 8.

So the latus rectum of E1 is 2b2a = 162 3 = 8 3, and that of E2 is 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

The intersection points satisfy x212 + y28 = 1 and x26 + y29 = 1, i.e.

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)

Keep practising — Conic Sections

Ques. 2JEE Main 2025, 29 January (Morning Shift)

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
Ques. 3JEE Main 2025, 29 January (Morning Shift)

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
Ques. 4JEE Main 2025, 29 January (Evening Shift)

If x + y = 109 is the equation of the chord of the ellipse x29 + y24 = 1, whose mid point is ( 52, 12 ). then + is equal to :

  1. 37
  2. 46
  3. 72
  4. 58
Ques. 5JEE Main 2025, 29 January (Evening Shift)

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C, whose mid-point is (1, 2), is:

  1. 4 2
  2. 2 2
  3. 2 3
  4. 3
Ques. 6JEE Main 2025, 29 January (Evening Shift)

Let y2=12x be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ)= 1474. Let C be the circle described taking PQ as a diameter. If the equation of a circle C is 64x2+64y2- x-64 3 y= , then - is equal to .

Ques. 7JEE Main 2025, 28 January (Morning Shift)

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
Ques. 8JEE Main 2025, 28 January (Morning Shift)

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 )
Ques. 9JEE Main 2025, 28 January (Morning Shift)

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 .

Ques. 10JEE Main 2025, 28 January (Evening Shift)

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
Ques. 11JEE Main 2025, 28 January (Evening Shift)

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

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