The tangent at any point on the curve x = a_{${\cos }^3\theta ,$} y=a_{${\sin }^3\theta$} meets the axes in P and Q. The locus of the mid point of PQ is

We have, _{$\frac{dy}{dx}$} =_{$\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}=$ $\frac{3a{\sin }^2\theta \cos \theta }{3a{\cos }^2\theta (-\sin \theta )}=-\tan \theta$}

Equation of tangent at _{$\text{'}\theta \text{'}$}  is _{$y-a{\sin }^3\theta$}  =-tan_$\theta$ (_{$x-a{\cos }^3\theta$} )

_{$\Rightarrow$  $\frac{x}{a\cos \theta }+\frac{y}{a\sin \theta }=1$}

_$\therefore$  P is _{$(a\cos \theta ,0)and(0,a\sin \theta ).$}  If mid point of PQ is (h, k), then

_{$h=\frac{a\cos \theta }{2},k=\frac{a\sin \theta }{2}$}  Eliminating _$\theta$ , we get

_{$h^2+k^2=\frac{a^2}{4}$}

_$\therefore$  focus is _{$x^2+y^2=\frac{a^2}{4}$}