JEE Main · 2024 · 04 Apr 2024 · Shift 2 · Hyperbola

Consider a hyperbola

H

having centre at the origin and foci on the x-axis. Let

C_{1}

be the circle touching the hyperbola

H

and having the centre at the origin. Let

C_{2}

be the circle touching the hyperbola

H

at its vertex and having the centre at one of its foci. If the areas (in sq units) of

C_{1}

and

C_{2}

are

36 π

and

4 π

respectively, then the length (in units) of latus rectum of

H

Answer: (B)

Solution

Hyperbola

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

C_{1}

(centre O, touching the hyperbola at its vertex) has radius

a

; area

36 π \Rightarrow a^{2} = 36, a = 6

C_{2}

is centred at a focus and touches the hyperbola at its vertex, so its radius is the focus-to-vertex distance

a (e - 1)

; area

4 π \Rightarrow a^{2} (e - 1)^{2} = 4 \Rightarrow 6 (e - 1) = 2 \Rightarrow e = \frac{4}{3}

(the alternative

a (e + 1) = 2

gives

e < 1

, impossible for a hyperbola).

b^{2} = a^{2} (e^{2} - 1) = 36 (\frac{16}{9} - 1) = 36 \times \frac{7}{9} = 28

.

Latus rectum

= \frac{2 b ^{2}}{a} = \frac{56}{6} = \frac{28}{3}

JEE Main · 2024 · 04 Apr 2024 · Shift 2 · Single correct option