Question

Tangent at a point $P$ on $\frac{ x ^{2}}{ a ^{2}}+\frac{ y ^{2}}{ b ^{2}}=1$ meets the $x$-axis at $A$ and $y$-axis at $B$. The locus of the midpoint of $A B$ is $\frac{a^{2}}{x^{2}}+\frac{b^{2}}{y^{2}}=k$, then find $k$.

Answer 1

$\frac{x \cos \theta}{a}+\frac{y \sin \theta}{b}=1$
$h =\frac{\frac{ a }{\cos \theta}+0}{2}, k =\frac{0+\frac{ b }{\sin \theta}}{2}$
$\cos \theta=\frac{ a }{2 h }, \sin \theta=\frac{ b }{2 k }$
$\cos ^{2} \theta+\sin ^{2} \theta=1$
$\frac{ a ^{2}}{4 h ^{2}}+\frac{ b ^{2}}{4 k ^{2}}=1$
$\therefore$ locus $\frac{ a ^{2}}{ x ^{2}}+\frac{ b ^{2}}{ y ^{2}}=4$