Locus Involving Argument

Locus Involving Argument

## Formula

$$\arg(z-z_0)=\theta$$

represents a ray starting from $z_0$ making an angle $$\theta$$ with the positive real axis.

For two fixed complex numbers $z_1,z_2$:

$$\arg\left(\frac{z-z_1}{z-z_2}\right)=\theta$$

represents the locus of points from which the line segment joining $z_1$ and $z_2$ subtends a constant angle $$\theta$$.

## Conditions / Special Cases

For:

$$\arg\left(\frac{z-z_1}{z-z_2}\right)=0$$

the points lie on the line through $z_1$ and $z_2$, excluding points where the expression is undefined.

## Usage

- Used to identify loci involving fixed angles and arguments of complex expressions.