Hyperbola
Circles Related to Hyperbola — Latus Rectum
nta_pyq_2024_apr
Grade 11
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
$\dfrac{14}{3}$
$\dfrac{28}{3}$
$\dfrac{11}{3}$
$\dfrac{10}{3}$
Step-by-Step Solution
Key Concept: Let $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. $C_1$ has centre at origin and touches $H$, so its radius $=a$ (touches at vertex). Area $=\pi a^2=36\pi\Rightarrow a=6$. $C_2$ touches $H$ at vertex with centre at focus: radius $=ae-a=a(e-1)$ or $a(e+1)$. Area $=\pi a^2(e-1)^2=4\pi$.
$a=6$, $e=4/3$, $b^2=28$. Length of latus rectum $=2b^2/a=28/3$.
Correct Answer: 2