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?
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 $$u^2+v^2+2u+v=0$$