# Complex Numbers

## Homework Statement

"The transformation T from the z-plane to the w-plane is given by

$w=\frac{1}{Z-2}$

where $Z=x+iy$ and $w=u+iv$

Show that under T the straight line with equation $2x+y=5$ is transformed to a circle in the w-plane with centre $\left ( 1,-\frac{1}{2} \right )$ and radius $\frac{\sqrt{5}}{2}$

## The Attempt at a Solution

I've worked out that the line $2x+y=5$ can be written in locus form as $\left|Z-10\right|=\left|Z+10-10i\right|$

##2x+y=5 \implies y=5-2x## so we're looking for the transform of ##z = x+i(5-2x)##.

##2x+y=5 \implies y=5-2x## so we're looking for the transform of ##z = x+i(5-2x)##.

okay I substituted z into the transformation but I cannot get an equation of a circle to come out, where do I go from here?

Fredrik
Staff Emeritus