Finding $a$ such that $ \sqrt{\frac32x^2-xy+\frac32y^2}=x\cos a+y\sin a$ has at least one solution other than $(0,0)$

Question

Find all values of the parameter $ a $ from the interval $ [0, 2 \pi) $, for which the equation $$ \sqrt{\dfrac{3}{2}x^2 - xy + \dfrac{3}{2}y^2} = x \cos a + y \sin a $$ has at least one solution $ (x, y) $ other than $ (0,0) $.

Answer 1

We know that $x\cos a+y\sin a\leq \sqrt{x^2+y^2}$ and so, $$ \sqrt{\dfrac{3}{2}x^2 - xy + \dfrac{3}{2}y^2} \leq\sqrt{x^2+y^2}\implies (x-y)^2\leq0\implies x=y$$

Now its trivial from here imho.