Consider a hyperbola H having centre at the origin and foci on the x-axis.

Question

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 areas (in sq units) of ${C}_{1}$ and ${C}_{2}$ are $36\pi $ and $4\pi $, respectively, then the length (in units) of latus rectum of $H$ is

Accepted Answer

$$\begin{aligned}
& C_1: x^2+y^2=a^2 \Rightarrow \text { area }=\pi a^2=36 \pi \Rightarrow a=6 \
& C_2: x^2+(y-a e)^2=(a e-a)^2 \
& \therefore \quad \pi(a e-a)^2=4 \pi \
& \Rightarrow 36(e-1)^2=4 \
& \Rightarrow e-1=\frac{1}{3} \Rightarrow e=\frac{4}{3} \
& \Rightarrow b^2=28 \
& \therefore \quad L R=\frac{2 b^2}{a}=\frac{2 \times 28}{6}=\frac{28}{3} \text { units }
\end{aligned}$$

Solutions · 1