Question

How to find the local of points representing z in the complex plane?

Can someone please explain to me how to do question 8? Thanks heaps!

Answer 1

The locus is the unit circle less the point `z=1`

Explanation:

We can split `z` as `z = x+yi` where `x` and `y` are real.

Then:

`(z+1)/(z-1) = (x+1+yi)/(x-1+yi)`

`color(white)((z+1)/(z-1)) = (x+1+yi)/(x-1+yi)`

`color(white)((z+1)/(z-1)) = ((x+1+yi)(x-1-yi))/((x-1+yi)(x-1-yi))`

`color(white)((z+1)/(z-1)) = (x^2-(1+yi)^2)/((x-1)^2-(yi)^2)`

`color(white)((z+1)/(z-1)) = ((x^2+y^2-1)-2yi)/(x^2-2x+1+y^2)`

If the real part is `0` then:

`x^2+y^2-1 = 0`

which is the equation of the unit circle.

Note that the actual locus excludes the point `z=1`, i.e. `(1, 0)`, since that would make the denominator of `(z+1)/(z-1)` zero.

graph{x^2+y^2=1 [-2.5, 2.5, -1.25, 1.25]}