Circles
Largest Circle Inside Ellipse at Given Centre
nta_pyq_2023_apr
Grade 11
Question:
If the radius of the largest circle with centre $(2,0)$ inscribed in the ellipse $x^2+4y^2=36$ is $r$, then $12r^2$ is equal to
Step-by-Step Solution
Key Concept: The normal to the ellipse $\frac{x^2}{36}+\frac{y^2}{9}=1$ at point $(6\cos\theta,3\sin\theta)$ must pass through $(2,0)$. Use the normal equation to find $\theta$.
$\cos\theta=\frac{4}{9}$, $\sin\theta=\frac{\sqrt{65}}{9}$. $r^2=(2-\frac{8}{3})^2+(\frac{\sqrt{65}}{3})^2=\frac{69}{9}$. $12r^2=92$.
Correct Answer: 2