Question

If any tangent to $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ intercepts lengths $h$ and $k$ on the axes, then $\frac{a^2}{h^2}+\frac{b^2}{k^2}=$

Answer 1

Let the point $P=(a\cos \theta ,b\sin \theta )$

Tangent is$\frac{x\times a\cos \theta }{a^2}+\frac{y\times b\sin \theta }{b^2}=1$

$\Rightarrow \frac{x}{a\sec \theta }+\frac{y}{b\csc \theta }=1$

$h=a\sec \theta$

$k=b\csc \theta$

$h^2=a^2{\sec }^2\theta$

$k^2=b^2{\csc }^2\theta$

$\Rightarrow \frac{a^2}{h^2}+\frac{b^2}{k^2}=\frac{1}{{\sec }^2\theta }+\frac{1}{{\csc }^2\theta }={\cos }^2\theta +{\sin }^2\theta =1$