JEE Main · 2022 · 28 Jul 2022 · Shift 2 · Parabola

Let the hyperbola

H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1

pass through the point

(22, - 22)

. A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?

(A) $(23, 32)$
(B) $(33, - 62)$ ✓ Correct
(C) $(3, - 6)$
(D) $(36, 62)$

Answer: (B)

Solution

The hyperbola has foci at

(\pm a e, 0)

. The parabola has focus at

(a e, 0)

and directrix passing through

(- a e, 0)

, so the vertex is at the origin and the semi-latus rectum is

2 a e

. Given that

4 a e = e \cdot \frac{2 b ^{2}}{a}

, we get

b^{2} = 2 a^{2}

. Since

(22, - 22)

lies on H, substituting gives

\frac{8}{a ^{2}} - \frac{8}{b ^{2}} = 1

, which simplifies to

\frac{1}{a ^{2}} - \frac{1}{b ^{2}} = \frac{1}{8}

. Solving with

b^{2} = 2 a^{2}

yields

a^{2} = 4

and

b^{2} = 8

. The eccentricity is

e = 1 + \frac{b ^{2}}{a ^{2}} = 3

. The parabola equation is

y^{2} = 4 (2 a e) x = 83 x

. Checking option (B):

(33)^{2} = 27

and

83 \cdot 33 = 72

, but

(- 62)^{2} = 72

, so it satisfies the equation.

JEE Main · 2022 · 28 Jul 2022 · Shift 2 · Single correct option

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JEE Main · 2022 · 28 Jul 2022 · Shift 2 · Parabola

Solution

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