How to solve a two variable equation

Question

Suppose $m,n$ are positive constants,there is an equation $m|\lambda_1|^2+|\lambda_2|^2-n\lambda_1\bar{\lambda_2}-n\lambda_2\bar{\lambda_1}=0$,where $\lambda_1,\lambda_2$ are nonzero complex numbers.

Does there exist solutions to this equation?

Answer 1

Let $\lambda_1 = |\lambda_1| e^{i \phi_1}$, and $\lambda_2 = |\lambda_2| e^{i \phi_2}$.

First note that the term $-n\lambda_1\bar{\lambda_2}-n\lambda_2\bar{\lambda_1} = -2 n |\lambda_1||\lambda_2| \cos(\phi_1-\phi_2)$. So let $-2 n \cos(\phi_1-\phi_2) = 2 q$, then we have

$$ 0 = m|\lambda_1|^2+|\lambda_2|^2+ 2 q |\lambda_1||\lambda_2|\\ = (|\lambda_2|+ q |\lambda_1|)^2 + (m-q^2)|\lambda_1|^2 $$

Solutions exist for $q^2 >m$ which is $n^2 \cos^2(\phi_1-\phi_2) > m$.