If p and q are lengths of perpendiculars from the origin on the tangent and the normal to the curve $x^{2/3}+y^{2/3}=a^{2/3},$ then  $4p^2+q^2=$

$$\begin{gathered} x=a{\cos }^3\theta ,y=a{\sin }^3\theta \\ \frac{dy}{dx}=\frac{-\sin \theta }{\cos \theta } \end{gathered}$$
Equation of tangent 
$$\begin{gathered} y-a{\sin }^3\theta =\frac{-\sin \theta }{\cos \theta }(x-a{\cos }^3\theta ) \\ x\sin \theta +y\cos \theta =a\cos \theta \sin \theta \end{gathered}$$
P=length of ${\perp }^{el}$ from (0,0) is 
$$\begin{gathered} \frac{\left|c\right|}{\sqrt{a^2+b^2}}=\frac{\left|a\sin \theta \cos \theta \right|}{\sqrt{{\sin }^2\theta +{\cos }^2\theta }} \\ P=\frac{1}{2}a.\sin 2\theta \end{gathered}$$
Equation of normal 
$$\begin{gathered} y-a{\sin }^3\theta =\frac{\cos \theta }{\sin \theta }(x-a{\cos }^3\theta ) \\ x\cos \theta -y\sin \theta =a\cos 2\theta \end{gathered}$$
Q=length of ${\perp }^{el}$from (0,0) is 
$$\begin{gathered} \frac{|-a\cos 2\theta |}{\sqrt{{\cos }^2\theta +{\sin }^2\theta }} \\ \therefore q=a\cos 2\theta \end{gathered}$$
$4p^2+q^2=\cancel{4}.\frac{1}{\cancel{4}}.a^2{\sin }^{2}2\theta +a^2{\cos }^{2}2\theta =a^2\left(1\right)=a^2$