13.0 Geometrical properties

1. Distance formulae: If ${z_1} = {x_1} + i{y_1}$ and ${z_2} = {x_2} + i{y_2}$ are two complex numbers, then the distance between them is $$\left| {{z_1} - {z_2}} \right| = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2}} $$

2. Section formulae: Segment joining points $A({z_1})$ and $B({z_2})$ is divided by point $P(z)$ in the ratio ${m_1}:{m_2}$, then $$z = \frac{{{m_1}{z_2} + {m_2}{z_1}}}{{{m_1} + {m_2}}}$$ where ${m_1}$ and ${m_2}$ are real.

3. Equation of straight line: The equation of straight line joining two points ${z_1}$ and ${z_2}$ is given by $$\frac{{z - {z_1}}}{{{z_2} - {z_1}}} = \frac{{\overline z - \overline {{z_1}} }}{{\overline {{z_2}} - \overline {{z_1}} }}$$ In determinant form, it is given by \[\left| {\begin{array}{c}z&{\overline z }&1 \\ {{z_1}}&{\overline {{z_1}} }&1 \\{{z_2}}&{\overline {{z_2}} }&1 \end{array}} \right| = 0\]

which is also the condition for three complex numbers $z$, ${z_1}$ and ${z_2}$ to be collinear.

The general equation of the straight line is $$\overline \alpha z + \alpha \overline z + r = 0$$ where $r$ is real and $\alpha $ is a non-zero complex constant.