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

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

1 Answer
Mar 1, 2018

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]}