Answer
The proof is below; $h_1=a$, and $h_2=b$.
Work Step by Step
If the two components are out of phase by $\pi/2$, one is a cosine function, and the other will be a sine function. We will call the amplitudes $h_1$ and $h_2$. Thus:
$\frac{x^2}{h_1^2}=cos^2 \omega t$
$\frac{y^2}{h_2^2}=sin^2 \omega t$
Thus, we see that it is an ellipse, and we see that $h_1=a$, and $h_2=b$.
