Question

If C is the centre and A, B are two points on the conic $4x^2+9y^2-8x-36y+4=0$ such that $\angle ACB=\frac{\pi }{2}$, then $C{A}^{-2}+C{B}^{-2}+\frac{23}{36}=$ ___

Answer 1

Given eqn is $4(x-1{)}^2+9(y-2{)}^2=36$ which is an ellipse C = (1,2)
Let CA makes an angle $\theta$ with the major axis
$A=(1+CAcos\theta ,2+CAsin\theta )$
$B=(1+CBcos[\frac{\pi }{2}+\theta ],2+CBsin[\frac{\pi }{2}+\theta ])$
A, B then on the ellipse
$\Rightarrow C{A}^2(4co{s}^2\theta +9si{n}^2\theta )=36$
$C{B}^2(4si{n}^2\theta +9co{s}^2\theta )=36$
$\Rightarrow C{A}^{-2}+C{B}^{-2}=\frac{13}{36}$