Explanation

(a) The Lissajous figure will be parabola if period ratio is $1:2$ and phase difference is $\pi/2$
Let $x = a\sin{(2\omega t + \pi/2)}\ and\ \ y = b\sin{\omega t}$
$\therefore\ \ \sin{\omega t} = y/b$
Now, $\frac{x}{a} = \sin{(2\omega t + \pi/2)} = \cos{2\omega/t}$
$= 1 - 2\sin^{2}{\omega t} = 1 - \frac{2y^{2}}{b^{2}}$
or $\frac{2y^{2}}{b^{2}} = 1 - \frac{x}{a} = - \left( \frac{x - a}{a} \right)$
or $y^{2} = - \frac{b^{2}}{2a}(x - a)$
it is an equation of a parabola as given in figure