complex numbers

You know

|z|*|z| = x*x+y*y

where z = x+jy, and j*j=-1,

and you can start from it.

on |z+2|=|z-1| square both sides--e.g.-(|a|)^2=(a)^2, then simplify to solve for z

Rap c∫/

Remember this implies

|x + iby + 2| = |x +iby - 1|

2 and 1 are real numbers so they plus your x coefficient must cancel leaving your complex component to always contra each other out. This is the critical insight to solve this equation, the real and imaginary components of z dodn't interact via an absolute function, so its your x component and -1 and +2 that are balancing, so you can ignore the iby component when your solving for x. So solve |x + 2| = |x - 1| then add i.(by) back in later.

Well now as |2| > |-1| it imples x must be negative so RHS will be positive and LHS negative before the Absolute takes effect. So lets take away the absolute and counter this by reversing the sign of the LHS, giving us:

-(x + 2) = x - 1,
-2x = -1 + 2 = 1
So x = -1/2

It kinda imples x = -1/2 doesn't it?

So if z = x + i.by, then for x = -1/2 and all b that equation holds true doesn't it?

|-1/2 + iby + 2| = |-1/2 + iby -1|
|+1.5 + iby| = |-1.5 + iby| for all b and y, where b and y are real numbers!

This is because the real and complex components don't interact so we can say

|+1.5 + iby| = 1.5 + i.|by| and
|-1.5 + iby| = 1.5 + i.|by|