Explanation

(b)
The equation of straight line touching the given circle is
$x\cos\theta + y\sin\theta = a$ …(i)
On differentiating $w.r.t.x,$ regarding $\theta$ as a constant
$\Rightarrow$ $\cos\theta + \frac{dy}{dx}\sin\theta = 0$ …(ii)
From eqs. (i) and (ii), we get
$\cos\theta = \frac{a\ \frac{dy}{dx}}{x\ \frac{dy}{dx} - y}$ and $\sin\theta = - \frac{a}{x\ \frac{dy}{dx} - y}$
$\therefore$ $\cos^{2}\theta + \sin^{2}\theta = 1$
$\therefore$ $\frac{a^{2}\left( \frac{dy}{dx} \right)^{2} + a^{2}}{{\left( x\ \frac{dy}{dx} - y \right)^{2}}_{\ }} = 1$
$\Rightarrow$ $\left( y - x\ \frac{dy}{dx} \right)^{2} = a^{2}\left\lbrack 1 + \left( \frac{dy}{dx} \right)^{2} \right\rbrack$