Find the locus of $w=1/z$ when $z$ lies on $y=2x+1$ For the transformation $w=\frac{1}{z}$, find the locus of $w$ when $z$ lies on the line with equation $y=2x+1$
I'm not quite sure how or where to start on this question. I know for instance that the locus of $z$ such that
$$|z-2| = |z+2-2i|$$
is the line $y=2x+1$. I'm not sure where I could go from here though
Thanks in advance

Suppose $z=x+iy$ then as $y=2x+1$ we have  $w=\dfrac{1}{x +(2x+1)i}$
$\dfrac{1}{x +(2x+1)i}=\dfrac{x-(2x+1)i}{5 x^2+4 x+1}$ 
If $w=a+bi$ then we have $(a,b)=\left(\dfrac{x}{5 x^2+4 x+1},\dfrac{-2 x-1}{5 x^2+4 x+1}\right)$
So 
$a=\dfrac{x}{5 x^2+4 x+1};\;b=\dfrac{-2 x-1}{5 x^2+4 x+1}$
$x=\dfrac{1-4a\pm\sqrt{-4 a^2-8 a+1}}{10 a}$
plugged in $b$ gives
$b=\dfrac{1}{2} \left(-1\pm\sqrt{1-4 a (a+2)}\right)$
$2b=-1\pm\sqrt{1-4 a (a+2)};\to (2b+1)^2=1-4 a (a+2)$
$a^2 + b^2+2a-b=0$
which is the equation of a circle. When $z$ varies on the blue line, $w$ varies on the orange circle
Hope it helps
remark
There are two points which are invariant 
$\left(-\dfrac{4}{5}, -\dfrac{3}{5}\right),( 0,1)$

