Explanation

(a)
Given equation can be rewritten as
$36\ \left( x^{2} - x + \frac{1}{4} \right) + 144\ \left( y^{2} - \frac{2}{3}y + \frac{1}{9} \right) = 144$
$\Rightarrow \frac{\left( x - \frac{1}{2} \right)^{2}}{4} + \frac{\left( y - \frac{1}{3} \right)^{2}}{1} = 1$
$\therefore\ e = \sqrt{1 - \frac{b^{2}}{a^{2}}} = \sqrt{1 - \frac{1}{4}} = \frac{\sqrt{3}}{2}$