Answer the following
(a) Find the Mobius Transformation that maps the line $Im z=1$ to the line $Rez=2$
(b) Find the Mobius Transformation that maps the line $Im z=1$ to the line $Rez=2,$ while mapping the point $i$ to the pount $2+10i$
(c) Find the Mobius Transformation that maps the circle $|z-1|=2$ to the circle $|z-3i|=4$
(d) Find the Mobius Transformation that maps the circle $|z-1|=2$ to the circle $|z-3i|=4,$ while mapping the point 3 to the point $-i.$
My attempt (a) I am able to transform the line $Im z=1$ to the line $Rez=2$ by rotation of $\pi/2$ translation of 2 unit along real axis. Similarly I could informally give the solution. How do I solve the problem formally? I could find the blinear transformation, if $z_1 \to w_1$, $z_2 \to w_2$ and $z_3 \to w_3$. using cross ratio. Can I use crossratio method here?