1. ## Complex Number

Let $\displaystyle z_1, z_2$ be nonzero complex numbers such that

$\displaystyle z_1^2 + z_2^2 = \sqrt{2}z_1 z_2$.

Show that $\displaystyle | z_1 | = | z_2 |$ and find $\displaystyle Arg{(\frac{z_1}{z_2})}$.

I've tried substituting $\displaystyle z = x+yi$ and $\displaystyle z = re^{i\theta}$ to prove this but it didnt work out. I assume there is a geometric or inequality approach to this question?

2. Originally Posted by shinn
Let $\displaystyle z_1, z_2$ be nonzero complex numbers such that

$\displaystyle z_1^2 + z_2^2 = \sqrt{2}z_1 z_2$.

Show that $\displaystyle | z_1 | = | z_2 |$ and find $\displaystyle Arg{(\frac{z_1}{z_2})}$.

I've tried substituting $\displaystyle z = x+yi$ and $\displaystyle z = re^{i\theta}$ to prove this but it didnt work out. I assume there is a geometric or inequality approach to this question?
In the equation $\displaystyle z_1^2 + z_2^2 = \sqrt{2}z_1 z_2$, divide through by $\displaystyle z_2^{\,2}$, and let $\displaystyle w = z_1/z_2$. You will get a quadratic equation for w.