## Cartesian Form of an Equation From the Complex Form

The equation
$| z - w \| = \|z-u \|$
is the complex form of a line. To get the Cartesian form of the line let
$z=x+iy, \: w=a+bi, \; u=c+di$
then
$| (x+iy)-(a+bi) \| = \|(x+iy)-(c+di) \|$

$| (x-a)+(y-b)i \| = \|(x-c)+(y-d)i \|$

$(x-a)^2+(y-b)^2 = (x-c)^2+(y-d)^2$

$x^2-2ax+a^2+y^2-2by+b^2 = x^2-2cx+c^2+y^2-2yd+d^2$

$-2ax+a^2-2by+b^2 =-2cx+c^2-2yd+d^2$

$a^2+2yd-2by+b^2 =2ax-2cx+c^2-2yd+d^2$

$2yd-2by =2ax-2cx-2yd-a^2-b^2+c^2+d^2$

$2y(d-b) =2(a-c)x-a^2-b^2+c^2+d^2$

$y =\frac{a-c}{d-b}x+ \frac{-a^2-b^2+c^2+d^2}{2(d-b)}$