Question

If $z_1,z_2$ are roots of equation $z^2-az+a^2=0$, then $\left|\frac{z_1}{z_2}\right|$ =

Answer 1

$z^2-az+a^2=0$

$z_1+z_2=a,z_1{z}_2=a^2$

$z_1{z}_2=(z_1+z_2{)}^2$

$z_1{z}_2=z_1^2+z_2^2+2z_1{z}_2$

$z_1^2+z_1^2+z_1{z}_2=0$

${\left(\frac{z_1}{z_2}\right)}^2+1+\left(\frac{z_1}{z_2}\right)=0$

$\frac{z_1}{z_2}=\frac{-1\pm i\sqrt{3}}{2}$

$\left|\frac{z_1}{z_2}\right|=\frac{1}{2}|\sqrt{1+3}|=\frac{2}{2}=1$