Question

Let $C$ be the largest circle centred at $(2,0)$ and inscribed in the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$If $(1, a)$ lies on $C$, then $10 \alpha^2$ is equal to

Answer 1

The correct answer is 118.

Equation of normal of ellipse 36x2​+16y2​=1 at any point P(6cosθ,4sinθ) is
3secθx−2cosecθy=10 this normal is also the normal of the circle passing through the point (2,0) So,
6secθ=10 or sinθ=0 (Not possible) cosθ=53​ and sinθ=54​ so point P=(518​,516​)
So the largest radius of circle
r=5320​​
So the equation of circle (x−2)2+y2=564​
Passing it through (1,α)
Then α2=559​
10α2=118