The transformation at $T$ given by $w=kz/(i+z)$ where $z\neq -i$, $k$ a real number, maps the complex number $2+i$ in the $z$-plane to its image $\frac{(3-i)}{2}$ in the $w$-plane.
a) Show that $k=2$
Point $P$ represents the complex number $z$ where $|z|=\sqrt{3}$. $T$ maps the point $P$ to point $Q$ in the w-plane.
b) Show that the locus of $Q$ is a circle with the cartesian equation given by: $(u-3)^2+v^2=3$ for $u, v\in\mathbb{R}$.
$T$ maps the point $z_0$ in the locus of $P$ to the point $w_0$ in the locus of $Q$, where the acute angle $\arg w_0$ is as large as possible.
c) Find the exact value of $|i+z_0|$
I did part (a) easily, but I couldn't do part (b). So far I did this: $$ iw=z(2-w)\Rightarrow z=iw/(2-w)\Rightarrow |z|=\left|\frac{iw}{2-w}\right|\Rightarrow \sqrt{3}=\frac{|iw|}{|(2-w)|}$$
Assume $z$=$x+iy$ then $|z|$=$x^2+y^2$=$3$
Now you apply the T on z.
Which gives $$w=\frac{2(x+iy)}{i+x+iy}$$ then make the denominator real by multiplying it with the complex conjugate. Then once you sorted the resulting equation into real and imaginary parts, equate u to the Real part and v to the imaginary part. Then substitute in the given equation $$(u-3)^2+v^2=3$$
Finally, evaluate the equation it should give you $x^2+y^2=3$ which is true. Hence QED.