Question

1.) Find the linear transformation w = f(z) such that the circle /z/ = 1 maps onto the circle /w-3+2i/=5 and f(-i) = 3 + 3i.
2). pls solve this: /z over (z+i)/ = 4

Answer 1

you want unit circle sent to circle with radius 5 centered at 3 - 2i rotated so that the bottom of the unit circle goes to the top of your target circle.
1) increase the radius by 5 so write 5z
2) rotate by pi, multiply by -1 get -5z
3) translate center get -5z+3-2i
so you should have
\[f(z)=-5z+3-2i\]
now we can check that
\[f(-i)=-5(-i)3+-2i=3+2i\] also notice that the left hand endpoint of the unit circle goes the the right hand endpoint of the new circle
\[f(-1)=5+3-2i=8+2i\] also
\[f(1)=-2-2i\] and
\[f(i)=3-7i\] sending the top of the unit circle to the bottom of your new circle

Answer 2

typo above. should be
\[f(-i)=-5(-i)+3-2i=3+3i\]

Answer 3

second on is easier. you have
\[\frac{z}{z+i}=4\]
\[z=4(z+i)=4+4i\]
\[a+bi=4a+4bi+4i=4a+(4b+4)i\]
two complex numbers are equal means real and imaginary parts are equal so you have
\[a=4a \iff a=0\]
\[b=4b+4\iff b=-\frac{4}{3}\]
so
\[z=-\frac{4}{3}i\]