Find the locus of $|\frac{z}{z-4}|=5$

Question

I have been asked to find the locus and the Cartesian equation of $\left|\frac{z}{z-4}\right|=5$.

I have tried several different ways and have got no easily sketchable graph. $|z|=5|z-4|$

I have tried expanding this and gotten nowhere, help would be very much appreciated.

Answer 1

Let $z=x+yi,$ where $x$ and $y$ are reals.

Thus, by your work we obtain: $$x^2+y^2=25((x-4)^2+y^2).$$ Can you end it now?

I got a circle $$\left(x-\frac{25}{6}\right)^2+y^2=\left(\frac{5}{6}\right)^2$$ without two points for $x=4$.