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Conic Sections

JEE Main 2025 (Online) 22nd January Morning ShiftMCQMedium📈 Topic in 29% papers+4−1
Ques. 1JEE 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

Answer

(4) 15

Solution

Circle C has radius 2 and touches both axes in the 2nd quadrant, so its centre is (-2,2).

[figure]

Distance between centres: d= 42+32=5

Two circles cut at exactly two points when

|r-2|<5<r+2

3<r<7 (dashed circles are the limiting tangent cases), so =3, =7

3 -2 =15

Answer: (4)

Keep practising — Conic Sections

Ques. 2JEE 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. 3JEE 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. 4JEE 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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