complex calc

please show steps on how to solve the problem

Problem 6, 15 Points. In this problem, we discuss the linear fractional transforma- tion T(2) = 92to " card, where a, b, c, de C and z = x + ly e CU {oo}. i) What is the condition in terms of a, b, c, d one have to impose so that T(z) is NOT a constant map? (ii) Consider the equation A(x2 + y?) + Br + Cy + D = 0 where r, y, A, B, C, D e R. If B2 + (2 -4AD > 0, show that this formula represents a circle if A # 0; show that this formula represents a line if A = 0. (iii) Consider the map W(z) = 1, z = r+iye CUjoo}. Show that the map W transforms circles and lines into circles and lines. Note that you are NOT required to prove that it maps circles to circles, or lines to lines. (iv) Show that the linear fractional transformation with your condition in (i) also trans- forms circles and lines into circles and lines, provided that any linear transformation U(2) = At + B. A, B e C maps circles to circles, and lines to lines