Explanation

(b)
Let the point on the $x$-axis be $(h,0)$.
The perpendicular distance from $(h,\ 0)$ to the line
$= \frac{\left| \frac{h}{a} - 1 \right|\ \ }{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}}}} = a\ \lbrack given\rbrack$
$\Rightarrow \frac{h}{a} - 1 = \pm a\frac{\sqrt{a^{2} + b^{2}}}{ab}$
$\Rightarrow h - a = \pm \frac{a}{b}\sqrt{a^{2} + b^{2}}$
$\Rightarrow h = a \pm \frac{a}{b}\sqrt{a^{2} + b^{2}}$
$= \frac{a}{b}\left( b \pm \ \sqrt{a^{2} + b^{2}} \right)$
$\therefore Required\ point\ is\ \left( \frac{a}{b}\left( b \pm \sqrt{a^{2} + b^{2}} \right),\ 0 \right)$