# Finding the image of this line under 1/z

Let $$T(z)=1/z$$. Find the image of $$y=2x+1$$ under $$T$$.

I assume x and y are the real and imaginary part of z.

Basically what I need was let $$w=1/(x+(2x+1)i)$$ then multiplied by the complex conjugate we have that the real imaginary part of x.

We have

$$w=\frac{x}{x^2+(2x+1)^2}+\frac{(2x+1)}{x^2+(2x+1)^2}i$$

Then should we equate coefficient with the equation of a circle? I dont know how to proceed.

From university, I know that the image must be a circle or a line since the transformation is a Mobius transformation. How is an A level student meant to get this?

i just looked at another question. Do i need to find two points such that this line is the locus for those two points? Then i sub that in?

• Did I edit your post correctly? – Arbuja Apr 29 '17 at 20:38

Let $w=\frac 1z\implies z=\frac 1w$
We write $w=u+iv$, so that $$z=x+iy=\frac{1}{u+iv}=\frac{u-iv}{u^2+v^2}$$
Then $$y=2x+1\implies -\frac{v}{u^2+v^2}=\frac{2u}{u^2+v^2}+1$$
This is a circle $$u^2+v^2+2u+v=0$$
• [+1] You can proceed by equivalence, i.e., replace the $\implies$ arrow by an $\iff$... – Jean Marie Apr 29 '17 at 21:06