# Geometry with Complex Numbers

## Geometry with Complex Numbers

Section Formula: If P(z) divides the line segment joining A(z₁) and B(z₂) internally in the ratio m:n, then $$z=\frac{m{{z}_{2}}+n{{z}_{1}}}{m+n}$$.

$$\frac{AP}{PB}=\frac{m}{n}$$,

If the division is external, then $$z=\frac{m{{z}_{2}}-n{{z}_{1}}}{m-n}$$.

Explanation: Let z₁ = x₁ + iy₁, z₂ = x₂ + iy₂. Then, A Ξ (x₁, y₁) and B Ξ (x₂, y₂). Let z = x + iy. Then P Ξ (x, y). We have from coordinate geometry.

$$x=\frac{m{{x}_{2}}+n{{x}_{1}}}{m+n}$$ and $$y=\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n}$$,

Hence, complex number of P is

$$z=\frac{m{{x}_{2}}+n{{x}_{1}}}{m+n}+i\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n}$$,

$$=\frac{m\left( {{x}_{2}}+i{{y}_{2}} \right)+n\left( {{x}_{1}}+i{{y}_{1}} \right)}{m+n}$$,

$$=\frac{m{{z}_{2}}+n{{z}_{1}}}{m+n}$$,

Distance of a given point from a given line

Let the given line be $$z\overline{a}+\overline{z}a+b=0$$ and the given point be zc. Then,

zc = xc + iyc

Replacing z by x + iy in the given equation, we get $$x\left( a+\overline{a} \right)+iy\left( \overline{a}-a \right)+b=0$$.

Distance of (xc, yc) from this line is

$$\frac{|{{x}_{c}}\left( a+\overline{a} \right)+i{{y}_{c}}\left( \overline{a}-a \right)+b|}{\sqrt{{{\left( a+\overline{a} \right)}^{2}}-{{\left( a-\overline{a} \right)}^{2}}}}=\frac{|{{z}_{c}}\overline{a}+{{\overline{z}}_{c}}a+b|}{\sqrt{4{{\left( \operatorname{Re}\left( a \right) \right)}^{2}}+4{{\left( \operatorname{Im}\left( a \right) \right)}^{2}}}}=\frac{|{{z}_{c}}\overline{a}+{{\overline{z}}_{c}}a+b|}{2|a|}$$.