Definition: 

Bilinear transformation: The transformation $w=\dfrac{az+b}{cz+d}$ where a,b,c,d are complex constants and $ad-bc\neq0$ is called as bilineat transformation.

Theorem:

Let the bilinear transformation be $w=\dfrac{az+b}{cz+d}$where a,b,c,d are complex constants and $ad-bc\neq0$

Dividing $az+b \ by\ \ cz+d$, we get

$w=\dfrac{a}{c}+\dfrac{b-\dfrac{ad}{c}}{cz+d}$

$w=\dfrac{a}{c}+\dfrac{b-ad}{c}\times\dfrac{1}{c(z+d/c)}$

$w=\dfrac{a}{c}+\dfrac{b-ad}{c^2}\times\dfrac{1}{\bigg(z+\dfrac{d}{c}\bigg)}$

Let $w_1=z+\dfrac {d}{c}$ which is a translation.

In translation, shape and size are preserved.

Hence the circle in z-plane is mapped in the circle in $w_1 $ plane

Let $w_2=\dfrac{1}{z+\dfrac{d}{c}}$$=\dfrac{1}{w_1}$, which is an inversion and reflection. In inversion the circle in $w_1$-plane is mapped in the circle in $w_2$  plane Let $w_3=\dfrac{bc-ad}{c^2}\times\dfrac{1}{z+\dfrac{d}{c}}=K_1.w_1$which is a rotation and magnification In rotation and magnification,the circle in  $w_2$-plane is mapped in the circle in  $w_3$  plane Let, $w_4=\dfrac{a}{c}+\dfrac{bc-ad}{c^2}\times\dfrac{1}{z+\dfrac{d}{c}}$$=K_2+w_3$which is a translation In translation, shape and size are preserved. Hence the circle in $w_3$-plane is mapped in the circle in $w_4$ plane.

Hence in bilinear transformation,the circle in z-plane is mapped in the circle in w-plane.