$x = a\cos \theta ,y = b\cos \theta$

Here, $x = a\cos \theta$ ...(i)
and $y = b\cos \theta$ ...(ii)
Differentiating (i) \& (ii) w.r.t. 0, we get

$\cfrac{{dx}}{{d\theta }} = a( - \sin \theta ) = - a\sin \theta$ and $\cfrac{{dy}}{{d\theta }} = b( - \sin \theta ) = - b\sin \theta$

therefore, $\cfrac{{dy}}{{dx}} = \cfrac{{dy/d\theta }}{{dx/d\theta }} = \cfrac{{ - b\sin \theta }}{{ - a\sin \theta }} = \cfrac{b}{a}$