# Locus of a complex number

### From above the locus of complex number z represents a circle with the center at ( 0, 0 ) and radius r = 2.

 EXAMPLE: Find the locus of a complex number z = x + iy, such that for a fixed point  $\fn_cm&space;\small&space;z_1=&space;x_1&space;+&space;i&space;y_1$ ,  |z-z1 |= aSolution:
##### which is a circle with center at $\fn_cm&space;\small&space;z_1=&space;(x_1,&space;y_1)$ and radius r = a
EXAMPLE: Let $\fn_cm&space;\small&space;z_1=&space;x_1&space;+&space;i&space;y_1$ be a fixed complex number. Find the locus of all complex number z = x + iy such that Arg(z-z1) = $\fn_cm&space;\small&space;\frac{\pi}{4}$Solution:
##### First we find the argument of z

so   $\fn_cm&space;\small&space;\theta&space;=&space;Arg(z-z_1)$

⇒  $\fn_cm&space;\small&space;Tan^{-1}\frac{y-y_1}{x-x_1}$

Because  Arg(z-z1) = $\fn_cm&space;\small&space;\frac{\pi}{4}$

So               $\fn_cm&space;\small&space;Tan^{-1}\frac{y-y_1}{x-x_1}$ = $\fn_cm&space;\small&space;\frac{\pi}{4}$

And            $\fn_cm&space;\small&space;\frac{y-y_1}{x-x_1}=&space;Tan&space;\frac{\pi}{4}$   = 1

So,  x-x1= y-y1       or    x – y =  x1-y1