Conic Sections

JEE Main 2024 (Online) 27th January Morning ShiftMCQMedium📈 Topic in 17% papers+4−1

Ques. 1JEE 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 :

Answer

(3) 20

Solution

The shortest distance from C to the parabola is measured along the normal at the nearest point, so that normal must pass through the centre C(2, 8).

Normal to y2 = 4x (a = 1) in slope form: y = mx - 2m - m3

Since it passes through (2, 8): 8 = 2m - 2m - m3, so m3 = -8 and m = -2 (the only real normal from C).

Its foot is P(am2, -2am) = (4, 4), hence

d2 = CP2 = (4 - 2)2 + (4 - 8)2 = 4 + 16 = 20

Answer: (3)

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Ques. 2JEE 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 :

16915
20095
15415
17415

Ques. 3JEE 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. 4JEE 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 :

8 53
3 5
4 5
10 53

Ques. 5JEE 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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