Find maximum of P

Question

For $a \ge0$ find maximum of $P=$$3x\sqrt{a-y^2}-3y\sqrt{a-x^2}+4xy+4\sqrt{a^2-ax^2-ay^2+x^2y^2}$

I think maximum of P when x=-y but i don’t know how to make it reasonable

Answer 1

Let $x=\sqrt{a}\cos\alpha$ and $y=\sqrt{a}\cos\beta,$ where $\{\alpha,\beta\}\subset[0,\pi]$.

Thus, by C-S $$P=a(3\sin(\beta-\alpha)+4\cos(\beta-\alpha))\leq a\sqrt{(3^2+4^2)(\sin^2(\beta-\alpha)+\cos^2(\beta-\alpha))}=5a.$$