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Conic Sections · Question 87

JEE Main 2025 (Online) 28th January Morning ShiftMCQMedium📈 Topic in 48% papers+4−1
1

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

Answer

(4) 754

Solution

(1, 0) is the focus of y2 = 4x (a = 1), so AC is a focal chord.

Let A = (t2, 2t); then C = ( 1t2, - 2t ) and length of focal chord = (t + 1t )2

(t + 1t )2 = 254 t + 1t = 52 t = 2 (or 12)

A = (4, 4), C = ( 14, -1 )

AD y-axis D = (4, -4); BC y-axis B = ( 14, 1 )

Parallel sides: AD = 8, BC = 2; distance between them = 4 - 14 = 154

Area = 12(8 + 2) 154 = 754

Answer: (4)

JEE Main 2025 (Online) 28th January Morning Shift

Keep practising — Conic Sections

2

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 )
JEE Main 2025 (Online) 28th January Morning Shift
3

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
4

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
5

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
6

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 .

JEE Main 2025 (Online) 28th January Evening Shift

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