Problems

Age
Difficulty
Found: 4

Prove the equalities:

a) \(\overline{z+w} = \overline{z} + \overline{w}\); b) \(\overline{zw} = \overline{z} \overline{w}\); c) \(\overline{\frac{z}{w}} = \frac{\overline{z}}{\overline{w}}\); d) \(|\overline{z}| = |z|\); d) \(\overline{\overline{z}} = z\).

Prove the equalities:

a) \(z + \overline {z} = 2 \operatorname{Re} z\);

b) \(z - \overline {z} = 2i \operatorname{Im} z\);

c) \(\overline {z} z = |z|^2\).

Let \(z_1\) and \(z_2\) be fixed points of a complex plane. Give a geometric description of the sets of all points \(z\) that satisfy the conditions:

a) \(\operatorname{arg} \frac{z - z_1}{z - z_2} = 0\);

b) \(\operatorname{arg} \frac{z_1 - z}{z - z_2} = 0\).