a)
\(
z_1 = 1 - i \\
z_2 = e^{ \frac{3}{4} \pi i} \\
z_2 = \cos(\frac{3}{4} \pi ) + i \sin(\frac{3}{4} \pi ) \\
z_2 = -\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \\
z_2 = -a + ia \\
\frac{z_2}{z_2 + z_1} = \frac{-a + ia}{-a + ia + 1 - i} =\\
\frac{-a + ia}{(1 - a) + i(a - 1)} =| \cdot \frac{(-a - ia) + (1 + i)}{(1 - a) - i(a - 1)} \\
\frac{a^2 + a^2 - a - ia + ia - a} {1 - 2a + a^2 + a^2 -2a + 1} = \frac{2a^2 -2a} {2a^2 -4a + 2} =\\
\\
\frac{ 2\cdot\frac{1}{2} - 2\frac{ \sqrt{2}} {2} } { 2\cdot\frac{1}{2}-4\frac{\sqrt{2}}{2}+2} = \frac{1-\sqrt{2}}{3-2\sqrt{2}} = \\
-1 -\sqrt{2} \\
\)
b)
\(
z = x + iy \\
\bar{z} = x - iy \\
z \cdot \bar{z} = (x + iy)(x - iy) = x^2 + y^2 = |z|^2
\)