Maximum value of $z_1^2 + z_2^2$ subject to conditions.

Question

Let $z_1$ and $z_2$ be two complex numbers such that $$\begin{eqnarray} z_1(z_1^2-3z_2^2)&=&2\tag{i}\\ z_2(3z_1^2-z_2^2)&=&1\tag{ii} \end{eqnarray}$$

If it is given that $k=z_1^2+z_2^2$ is a real number, then what is the value of $k$?

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(I have absolutely no idea how to proceed.)

Answer 1

HINT:

$$2+i=z_1^3+3z_1^2(iz_2)+3z_1(iz_2)^2+(iz_2)^3=(z_1+iz_2)^3$$

$$ 2-i=?$$

$$(2-i)(2+i)=?$$