Explanation

(c)
We have, $4x^{2} + 16y^{2} - 24x - 32y = 1$
$\Rightarrow 4\left( x^{2} - 6x \right) + 16\left( y^{2} - 2y \right) = 1$
$\Rightarrow 4\left( x^{2} - 6x + 9 \right) + 16\left( y^{2} - 2y + 1 \right) - 36 - 16 = 1$
$\Rightarrow 4{(x - 3)}^{2} + 16{(y - 1)}^{2} = 53$
$\Rightarrow \frac{{(x - 3)}^{2}}{\frac{53}{4}} + \frac{{(y - 1)}^{2}}{\frac{53}{16}} = 1$
On comparing with $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$, we get
$a^{2} = \frac{53}{4}$ and $b^{2} = \frac{53}{16}$
$\therefore$ Eccentricity of ellipse is $e = \sqrt{\frac{a^{2} - b^{2}}{a^{2}}}$
$\Rightarrow e = \sqrt{\frac{\left( \frac{53}{4} \right) - \left( \frac{53}{16} \right)}{\left( \frac{53}{4} \right)}}$
$\Rightarrow e = \frac{\sqrt{3}}{2}$