Conic Sections

JEE Main 2024 (Online) 29th January Morning ShiftNumericalMedium📈 Topic in 29% papers+4

Ques. 1JEE Main 2024, 29 January (Morning Shift)

If the points of intersection of two distinct conics x2+y2=4b and x216+ y2b2=1 lie on the curve y2=3x2, then 3 3 times the area of the rectangle formed by the intersection points is .

Answer

432

Solution

Put y2=3x2 in the circle: 4x2=4b x2=b

In the ellipse: b16+ 3bb2=1 b2-16b+48=0 b=4 or 12

b = 4 gives the same circle x2+y2=16 twice (not distinct) ⇒ b = 12

Points: x2=12, y2=36 ⇒ ( 2 3, 6 )

Rectangle area =4 3 12=48 3 ⇒ 3 3 48 3=432

Answer: 432

Keep practising — Conic Sections

Ques. 2JEE Main 2024, 29 January (Evening Shift)

Let P( , ) be a point on the parabola y2=4x. If P also lies on the chord of the parabola x2=8y whose mid point is (1, 54 ), then ( -28)( -8) is equal to .

Ques. 3JEE Main 2024, 27 January (Morning Shift)

If the shortest distance of the parabola y2=4x from the centre of the circle x2+y2-4x-16y+64=0 is d, then d2 is equal to :

Ques. 4JEE Main 2024, 27 January (Morning Shift)

The length of the chord of the ellipse x225+ y216=1, whose mid point is (1, 25 ), is equal to :

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15415
17415

Ques. 5JEE Main 2024, 27 January (Morning Shift)

Four distinct points (2k,3k),(1,0),(0,1) and (0,0) lie on a circle for k equal to :

Ques. 6JEE Main 2024, 27 January (Evening Shift)

Let e1 be the eccentricity of the hyperbola x216- y29=1 and e2 be the eccentricity of the ellipse x2a2+ y2b2=1, a>b, which passes through the foci of the hyperbola. If e1e2=1, then the length of the chord of the ellipse parallel to the x-axis and passing through (0,2) is :

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3 5
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10 53

Ques. 7JEE Main 2024, 27 January (Evening Shift)

Consider a circle (x- )2+(y- )2=50, where , >0. If the circle touches the line y+x=0 at the point P, whose distance from the origin is 4 2, then ( + )2 is equal to .

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