The resultant of two rectangular simple harmonic motions of the same frequency and unequal amplitudes but differing in phase by $\frac{\mathrm{\pi }}{2}$ is

If first equation is ${\mathrm{y}}_1 = {\mathrm{a}}_1 \sin \mathrm{\omega t}$

$\Rightarrow \sin \mathrm{\omega t} = \frac{{\mathrm{y}}_1}{{\mathrm{a}}_1}$-------------------(i)

then second equation will be

${\mathrm{y}}_2 = {\mathrm{a}}_2 \sin (\mathrm{\omega t}+\frac{\mathrm{\pi }}{2})$

$= {\mathrm{a}}_2[\sin \mathrm{\omega t} \cos \frac{\mathrm{\pi }}{2}+\cos \mathrm{\omega tsin} \frac{\mathrm{\pi }}{2}] = {\mathrm{a}}_2 \cos \mathrm{\omega t}$

$\Rightarrow \cos \mathrm{\omega t} = \frac{{\mathrm{y}}_2}{{\mathrm{a}}_2}$--------------(ii)

By squaring and adding equations (i) and (ii)

${\sin }^2 \mathrm{\omega t} + {\cos }^2 \mathrm{\omega t} = \frac{{{\mathrm{y}}^2}_1}{{{\mathrm{a}}^2}_1}+\frac{{{\mathrm{y}}_2}^2}{{{\mathrm{a}}_2}^2}{ }^{ }$

$\Rightarrow \frac{{{\mathrm{y}}^2}_1}{{{\mathrm{a}}^2}_1}+\frac{{{\mathrm{y}}_2}^2}{{{\mathrm{a}}_2}^2}$= 1. This is the equation of ellipse.