Paper 1, Section II, A

Algebra and Geometry | Part IA, 2007

(i) Show that any line in the complex plane C\mathbb{C} can be represented in the form

cˉz+czˉ+r=0,\bar{c} z+c \bar{z}+r=0,

where cCc \in \mathbb{C} and rRr \in \mathbb{R}.

(ii) If zz and uu are two complex numbers for which


show that either zz or uu is real.

(iii) Show that any Möbius transformation

w=az+bcz+d(bcad0)w=\frac{a z+b}{c z+d} \quad(b c-a d \neq 0)

that maps the real axis z=zˉz=\bar{z} into the unit circle w=1|w|=1 can be expressed in the form

w=λz+kz+kˉw=\lambda \frac{z+k}{z+\bar{k}}

where λ,kC\lambda, k \in \mathbb{C} and λ=1|\lambda|=1.

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