Explanation

(a)
We have, $\sqrt{1 - x^{2}} + \sqrt{1 - y^{2}} = a(x - y)$
Putting $x = \sin A,\ y = \sin B$, we get
$\cos A + \cos B = a(\sin A - \sin B)$
$\Rightarrow$ $\cot\frac{A - B}{2} = a$
$\Rightarrow$ $A - B = 2\cot^{- 1}a$
$\Rightarrow$ $\sin^{- 1}x - \sin^{- 1}y = 2\cot^{- 1}a$
On differentiating $w.r.t.x$, we get
$\frac{1}{\sqrt{1 - x^{2}}} - \frac{1}{\sqrt{1 - y^{2}}}\ \frac{dy}{dx} = 0$
$\Rightarrow$ $\frac{dy}{dx} = \frac{\sqrt{1 - y^{2}}}{\sqrt{1 - x^{2}}}$
Clearly, it is differential equation of the first order and first degree.