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 $AB$ is $\frac{a^2}{x^2}+\frac{b^2}{y^2}=k$, then find $k$.

Answer 1

Answer: 4.00

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