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Conic Sections

JEE Main 2025 (Online) 23rd January Evening ShiftMCQMedium📈 Topic in 13% papers+4−1
Ques. 1JEE 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

Answer

(2) x2+y2-6x+5=0

Solution

Shortest distance from (a,0) is along the normal at T(t2,2t): a=2+t2, PT2=4+4t2=16 t2=3, a=5.

[figure]

Focus S(1,0) and (5,0) lie on the axis, and the centre is on the axis they are the ends of a diameter.

Circle: (x-1)(x-5)+y2=0 x2+y2-6x+5=0

Answer: (2)

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Ques. 2JEE 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. 3JEE 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
Ques. 4JEE Main 2025, 22 January (Morning Shift)

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2,5) and intersects the circle C at exactly two points. If the set of all possible values of r is the interval ( , ), then 3 -2 is equal to :

  1. 10
  2. 12
  3. 14
  4. 15
Ques. 5JEE Main 2025, 22 January (Morning Shift)

Let the parabola y=x2+px-3, meet the coordinate axes at the points P, Q and R. If the circle C with centre at (-1,-1) passes through the points P, Q and R, then the area of PQR is :

  1. 4
  2. 6
  3. 5
  4. 7
Ques. 6JEE Main 2025, 22 January (Evening Shift)

Let P(4,4 3) be a point on the parabola y2=4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :

  1. 34 33
  2. 343 38
  3. 17 3
  4. 263 38
Ques. 7JEE Main 2025, 22 January (Evening Shift)

Let E: x2a2+ y2b2=1, a>b and H: x2A2- y2B2=1. Let the distance between the foci of E and the foci of H be 2 3. If a-A=2, and the ratio of the eccentricities of E and H is 13, then the sum of the lengths of their latus rectums is equal to :

  1. 10
  2. 7
  3. 9
  4. 8

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