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Conic Sections

JEE Main 2025 (Online) 2nd April Morning ShiftNumericalMedium📈 Topic in 38% papers+4
Ques. 1JEE Main 2025, 2 April (Morning Shift)

The absolute difference between the squares of the radii of the two circles passing through the point (-9, 4) and touching the lines x + y = 3 and x - y = 3, is equal to .

Answer

768

Solution

The lines meet at (3, 0), and their angle bisectors are y = 0 and x = 3.

At (-9, 4), x + y - 3 = -8 < 0 and x - y - 3 = -16 < 0, so the point lies in the angle bisected by y = 0 (the part x < 3).

Hence the centre is (h, 0) with h < 3, and the radius is its distance from x + y = 3, i.e. r = 3 - h 2.

Since the circle passes through (-9, 4),

(h + 9)2 + 16 = (3 - h)22

2h2 + 36h + 194 = h2 - 6h + 9

h2 + 42h + 185 = 0 (h + 5)(h + 37) = 0

For h = -5, r12 = 642 = 32; for h = -37, r22 = 16002 = 800.

|r22 - r12| = 800 - 32 = 768

Answer: 768

Keep practising — Conic Sections

Ques. 2JEE Main 2025, 2 April (Evening Shift)

Let the point P of the focal chord PQ of the parabola y2 = 16x be (1, -4). If the focus of the parabola divides the chord PQ in the ratio m : n, (m, n) = 1, then m2 + n2 is equal to :

  1. 17
  2. 37
  3. 10
  4. 26
Ques. 3JEE Main 2025, 2 April (Evening Shift)

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :

  1. 3 19
  2. 316
  3. 4 17
  4. 57
Ques. 4JEE 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
Ques. 5JEE 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. 6JEE 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. 7JEE 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. 8JEE 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. 9JEE 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. 10JEE 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. 11JEE 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 )

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