Orthogonality in Hilbert space

Question

Let $H$ be a Hilbert space. I want to prove that if for all $x,y\in H$ and $\alpha\in \mathbb{K}$ $$\|x+\alpha y\|=\|x-\alpha y\|,$$ then $x\perp y$. I was able to show that Re$(x,y)=0$. Any ideas on how to show that Im$(x,y)=0$?

Answer 1

Hint: If $\alpha=1$, we have $$ \langle x+\alpha y, x+\alpha y\rangle = \|y\|^2 + 2 \Re\langle x,y \rangle + \|x\|^2\\ \langle x-\alpha y, x-\alpha y\rangle = \|y\|^2 - 2 \Re\langle x,y \rangle + \|x\|^2 $$ If $\alpha = i$, we have $$ \langle x+\alpha y, x+\alpha y\rangle = \|y\|^2 + 2 \Im\langle x,y \rangle + \|x\|^2\\ \langle x-\alpha y, x-\alpha y\rangle = \|y\|^2 - 2 \Im\langle x,y \rangle + \|x\|^2 $$