Search problems

2 problems match “modulus of complex numbers”. Clear

Visitors search a sample. Sign in for free to search every problem. Sign in

1. Pairs of unit complex numbers ●●○○○ 20 ptsFind the number of pairs $(z_1,z_2)$ of complex numbers such that $z_1 z_2$ is pure imaginary, $\frac{z_1}{z_2}$ is real, and $|z_1| = |z_2| = 1$.
2. Smallest modulus in a complex system ●●●○○ 50 ptsThe complex numbers $z$ and $w$ satisfy the system $$\begin{align*} z + \frac{20i}{w} &= 5+i, \\ w+\frac{12i}{z} &= -4+10i. \end{align*}$$ Find the smallest possible value of $|zw|^2$.