Explanation

(c)
Slope of $2y = x\ is\frac{1}{2}\left( m_{1},\ say \right)$
and slope of $3y + 4x = 0\ is - \frac{4}{3}\left( m_{2},say\ \right)$
$m_{1}m_{2} = - \frac{b^{2}}{a^{2}}$
$\Rightarrow \ \ \left( \frac{1}{2} \right)\left( - \frac{4}{3} \right) = - \frac{b^{2}}{a^{2}}$
$\Rightarrow \ \ \frac{b^{2}}{a^{2}} = \frac{2}{3}$
Eccentricity, $e = \sqrt{1 - \frac{b^{2}}{a^{2}}}$
$= \sqrt{1 - \frac{2}{3}} = \frac{1}{\sqrt{3}}$