Fractional transformation

Exercise2.2

1. What straight line or circle is the image of the next straight line or circle mapped by $w = \frac{1}{z}$.

(a)

A straight line that intersects the unit circle $\vert z\vert = 1$ at two points P and Q

(b)

A straight line tangent to the unit circle at one point P

(c)

A circle passing through 3 points $a$ (real number), $i$, $-i$

2. Find the invariant point by the following linear transformation on the complex plane.

(a)

$w = \frac{1}{z}$

(b)

$w = \frac{az + b}{cz + d} (ad -bc \neq 0)$