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Complex Number Pairs

Find 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 $ |z_1| = |z_2| = 1 $.

1
2
3
+
4
5
6
-
7
8
9
$\frac{a}{b}$
.
0
=
%

$a^n$
$a^{\circ}$
$a_n$
$\sqrt{}$
$\sqrt[n]{}$
$\pi$
$\ln{}$
$\log$
$\theta$
$\sin{}$
$\cos{}$
$\tan{}$

$($
$)$
$[$
$]$
$\cap$
$\cup$
$,$
$\infty$