Explanation

(a) Let the equations of two mutually perpendicular SHM’s same frequency be
$x = a_{1}\sin{\omega t}\ and\ y = a_{2}sin(\omega t + \varnothing)$
Then, the general equation of Lissajous figure can be obtained
as $\frac{x^{2}}{a_{1}^{2}} + \frac{y^{2}}{a_{2}^{2}} - \frac{2xy}{a_{1}a_{2}}\cos{\varnothing = \sin^{2}\varnothing}$
For $\varnothing = 0{^\circ}:\frac{x^{2}}{a_{1}^{2}} + \frac{y^{2}}{a_{2}^{2}} - \frac{2xy}{a_{1}a_{2}} = 0$
$\Rightarrow \ \ \ \ \left\lbrack \frac{x}{a_{1}} - \frac{y}{a_{2}} \right\rbrack^{2} = 0 \Rightarrow \frac{x}{a_{1}} = \frac{y}{a_{2}} \Rightarrow y = \frac{a_{2}}{a_{1}}x$
This is an equation of a straight line passing through origin.