Explanation

(a)
Given, $x = 3\ (\cos{t + \sin{t),\ y = 4(\cos{t - \sin t)\ }}}$
$\Rightarrow \ \ \frac{x}{3} = \cos{t + \sin{t,\ \frac{y}{4} = \cos{t - \sin t\ }}}$
$\therefore\left( \frac{x}{3} \right)^{2} + \left( \frac{y}{4} \right)^{2} = \left( \cos{t + \sin t} \right)^{2} + \left( \cos t - \sin t \right)^{2}$
$\Rightarrow \ \frac{x^{2}}{9} + \frac{y^{2}}{16} = 2$
$\Rightarrow \ \ \frac{x^{2}}{18} + \frac{y^{2}}{32} = 1$, which is an ellipse