Explanation

(a)
The line $y = mx + c$ touches the circle $x^{2} + y^{2} = r^{2}$, if and only if $c = \pm r\sqrt{1 + m^{2}}$ here we have line $3x - 2y = k$
$\Rightarrow \ \ \ y = \frac{3}{2}x - \frac{1}{2}k$
and circle $x^{2} + y^{2} = 4r^{2}$
$\therefore$ By condition $c = \pm a\sqrt{1 + m^{2}}$, we have
$- \frac{1}{2}k = \pm 2r\sqrt{1 + \frac{9}{4}}$
On squaring both sides, we get
$\frac{1}{4}k^{2} = 4r^{2}\left( \frac{13}{4} \right)$
$\Rightarrow \ \ \ k^{2} = 52r^{2}$