Explanation

(b)
Let $P(a{cos\ \theta}{b\ \sin\ \theta),\ \mathcal{Q}(a\cos\theta, - b\sin\theta)\ \ }$
Let a point $R(h,k)$ divides the line joining the points $P$ and $\mathcal{Q}$ internally in the ration 1:2
Then , $PR:R\mathcal{Q} = 1:2$
Now, by division formula
$h = a\cos\theta\ \Rightarrow \cos{\theta = \frac{h}{a}\ \ \ \ .\ ..(i)}$
and $k = \frac{b}{3}\sin\theta$
$\Rightarrow \ \sin{\theta = \frac{3k}{b}}$ … (ii)
On squaring and adding Eqs. (i) and (ii), we get
$\frac{h^{2}}{a^{2}} + \frac{9k^{2}}{b^{2}} = 1$
Hence locus of $R$ is
$\frac{x^{2}}{a^{2}} + \frac{9y^{2}}{b^{2}} = 1\$