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Conic Sections

JEE Main 2025 (Online) 24th January Morning ShiftMCQHard📈 Topic in 53% papers+4−1
Ques. 1JEE Main 2025, 24 January (Morning Shift)

Let the product of the focal distances of the point ( 3, 12 ) on the ellipse x2a2+ y2b2=1, (a>b), be 74. Then the absolute difference of the eccentricities of two such ellipses is

  1. 1-2 2 3
  2. 1- 3 2
  3. 3-2 22 3
  4. 3-2 23 2

Answer

(3) 3-2 22 3

Solution

Focal distances of P(x1,y1) are a ex1, so their product is a2-e2x12.

[figure]

a2-3e2= 74 4a2=7+12e2; P on the ellipse with b2=a2(1-e2):

12(1-e2)+1=4a2(1-e2) 12e4-17e2+6=0

e2= 23, 34, so |e1-e2|= 32- 23= 3-2 22 3

Answer: (3)

Keep practising — Conic Sections

Ques. 2JEE Main 2025, 24 January (Morning Shift)

Let circle C be the image of x2+y2-2x+4y-4=0 in the line 2x-3y+5=0 and A be the point on C such that OA is parallel to x-axis and A lies on the right hand side of the centre O of C. If B( , ), with <4, lies on C such that the length of the arc AB is (1/6)th of the perimeter of C, then - 3 is equal to

  1. 4- 3
  2. 3
  3. 4
  4. 3+ 3
Ques. 3JEE Main 2025, 24 January (Evening Shift)

If the equation of the parabola with vertex V ( 32,3 ) and the directrix x+2y=0 is x2+ y2- xy-30x-60y+225=0, then + + is equal to :

  1. 6
  2. 8
  3. 7
  4. 9
Ques. 4JEE Main 2025, 24 January (Evening Shift)

The equation of the chord, of the ellipse x225+ y216=1, whose mid-point is (3,1) is :

  1. 5x+16y=31
  2. 48x+25y=169
  3. 4x+122y=134
  4. 25x+101y=176
Ques. 5JEE Main 2025, 24 January (Evening Shift)

Let H1: x2a2- y2b2=1 and H2:- x2A2+ y2B2=1 be two hyperbolas having length of latus rectums 15 2 and 12 5 respectively. Let their eccentricities be e1= 52 and e2 respectively. If the product of the lengths of their transverse axes is 100 10, then 25e22 is equal to .

Ques. 6JEE Main 2025, 23 January (Morning Shift)

If the line 3x-2y+12=0 intersects the parabola 4y=3x2 at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to

  1. 2- -1 ( 32 )
  2. -1 ( 97 )
  3. -1 ( 119 )
  4. -1 ( 45 )
Ques. 7JEE Main 2025, 23 January (Morning Shift)

Let the circle C touch the line x-y+1=0, have the centre on the positive x-axis, and cut off a chord of length 4 13 along the line -3x+2y=1. Let H be the hyperbola x2 2- y2 2=1, whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C. Then 2 2+3 2 is equal to .

Ques. 8JEE Main 2025, 23 January (Evening Shift)

The length of the chord of the ellipse x24+ y22=1, whose mid-point is (1, 12 ), is :

  1. 23 15
  2. 13 15
  3. 15
  4. 53 15
Ques. 9JEE Main 2025, 23 January (Evening Shift)

Let the shortest distance from (a,0), a>0, to the parabola y2=4x be 4. Then the equation of the circle passing through the point (a,0) and the focus of the parabola, and having its centre on the axis of the parabola is :

  1. x2+y2-8x+7=0
  2. x2+y2-6x+5=0
  3. x2+y2-4x+3=0
  4. x2+y2-10x+9=0
Ques. 10JEE Main 2025, 23 January (Evening Shift)

The focus of the parabola y2=4x+16 is the centre of the circle C of radius 5. If the values of , for which C passes through the point of intersection of the lines 3x-y=0 and x+ y=4, are 1 and 2, 1< 2, then 12 1+29 2 is equal to .

Ques. 11JEE Main 2025, 22 January (Morning Shift)

Let the foci of a hyperbola be (1,14) and (1,-12). If it passes through the point (1,6), then the length of its latus-rectum is :

  1. 256
  2. 1445
  3. 2885
  4. 245

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