Explanation

(a)
The equation of tangent is
$\frac{x}{a}\sec{\theta - \frac{y}{b}\tan{\theta = 1}}$
$\therefore\ \$Coordinates of $A$ and $B$ are ($a\cos{\theta,\ 0)}\$and $(0,\ - b\cot{\theta)}$respectively.
Let coordinates of $P$ are$(h,\ k)$.
$\therefore\ \ h = a\cos{\theta,\ }k = - b\cot\theta$
$\Rightarrow \frac{k}{h} = - \frac{b}{a\sin\theta}$
$\Rightarrow \sin{\theta = \ } - \frac{bh}{ak}$
$\Rightarrow \frac{b^{2}h^{2}}{a^{2}k^{2}} = \sin^{2}\theta$
$\Rightarrow \ \frac{b^{2}h^{2}}{a^{2}k^{2}} + \frac{h^{2}}{a^{2}} = 1$
$\Rightarrow \ \frac{b^{2}}{k^{2}} + 1 = \frac{a^{2}}{h^{2}}$
$\Rightarrow \frac{a^{2}}{h^{2}} - \frac{b^{2}}{k^{2}} = 1$
Hence, the locus of $P$ is $\frac{a^{2}}{x^{2}} - \frac{b^{2}}{y^{2}} = 1$