If P1 and P2 be the lengths of perpendiculars from the origin on the tangent and normal to the curve $x^{2/3}+y^{2/3}=a^{2/3}$ respectively, the value of $4P_1^2+P_2^2$ is:

$a^2$

$x^{2/3}+y^{2/3}=a^{2/3}$. Let $P ≡ (a\cos^3θ, a\sin^3θ)$ be general point on curve.

Equation of tangent is : $y\cos θ +x \sin θ=\frac{a}{2}\sin 2θ$

Equation of normal is : $x\cos θ - y\sin θ = a\cos 2θ$

P₁ = ⊥ distance of tangent from origin = $\frac{|\frac{a}{2}\sin 2θ|}{\sqrt{\cos^θ+\sin^2θ}}=|\frac{a}{2}\sin 2θ|$ . . . . (i)

P₂ = ⊥ distance of normal from origin = $\frac{|a\cos 2θ|}{\sqrt{\cos^θ+\sin^2θ}}=|a\cos 2θ|$ . . . . (ii)

From (i) and (ii), $4P_1^2+P_2^2=a^2(\sin^22θ+\cos^22θ)=a^2$