Explanation

(c)
Since, $\overline{z} + i\overline{w} = 0\ \ \ \ \ \ \ \ \Rightarrow \ \ \ \ \ \ \ \overline{z} = - i\overline{w}$
$\Rightarrow \ \ \ \ \ \ z = - iw\ \ \ \ \ \ \Rightarrow \ \ \ \ \ \ w = - iz$
Also, $\arg{(zw) = \pi}\ \ \ \ \ \ \Rightarrow \ \ \ \ \ \arg{\left( - iz^{2} \right) = \pi}$
$\Rightarrow \ \ \ \ \ \arg{( - i) + 2\arg{(z)}} = \pi$
$\Rightarrow \ \ - \frac{\pi}{2} + 2\arg{(z) = \pi}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left\lbrack \because\ \arg{( - i) = - \frac{\pi}{2}} \right\rbrack$
$\Rightarrow \ \ \ \ \ 2\arg{(z) = \frac{3\pi}{2}}$
$\Rightarrow \ \ \ \ \arg{(z) = \frac{3\pi}{4}}$