Mobius transformations problem 
The Mobius transformation $T_1(z)=\dfrac{z+b}{z+d}$ maps the line $\operatorname{Im}(z)=\operatorname{Re}(z)+3$ onto the unit circle in such a way that the region above the line is mapped to the interior of the circle and that $5i$ is mapped to the origin. Find the value of $d$.


The Mobius transformation $T_2(z)=\dfrac{az+b}{z-1}$ maps unit circle onto the line $\operatorname{Re}(z)=3$ in such a way that the interior of the circle is mapped to the left of the line and that the origin gets mapped to $2+i$. Find $a$.


Let $T_1$ and $T_2$ be as in the previous two parts, and let $T_3(z)=T_2\circ T_1(z)$. Then we can consider $T_3$ to be the composition of a translation followed by a dilation followed by a rotation. By what factor does $T_3$ dilate?


I have already solved the first two problems (the answers are $-2-3i$ and $4+i$, respectively), but I'm not sure how to use the information ascertained from the solutions to solve the third and final problem. Any help (or maybe a solution) would be much appreciated!

Hint: Calculate $a,b,c,d$ where $T_3(z)=(az+b)/(cz+d)$.  
This is possible because the Mobius transformations form a group, with composition as the operation.   
I get $T_3(z)=1/2((1-i)z-1-3i)$.  The dilation is by $|a|=1/\sqrt2$.
