a-b-C-ab-b-0-f-z-az-b-such-that-f-M-M-1-let-z-A-z-and-z-A-z-and-f-A-A-show-that-2Re-b-z-bb-A-is-the-set-of-invariant-points-and-describes

Question Number 209707 by alcohol last updated on 19/Jul/24

a, b \in C : a \bar{b} + b = 0 f : z^{'} = a \bar{z} + b

s u c h t h a t f (M) = M^{'}

1 . l e t z_{A} = z a n d z_{A^{'}} = z^{'} a n d f (A) = A

s h o w t h a t 2 R e (\bar{b z}) = b \bar{b}

(A i s t h e s e t o f i n v a r i a n t p o i n t s a n d

d e s c r i b e s a l i n e (△))

2 . D e d u c e t h a t (△) i s a l i n e w i t h

g r a d i e n t \vec{u} w i t h a f f i x z_{\vec{u}} = i b

3 . s h o w t h a t \frac{z_{M M^{'}}}{z_{u}} = \frac{b \bar{b} - 2 R e (b \bar{z})}{i b \bar{b}}

4 . s h o w t h a t 2 R e ({\bar{b z}}_{0}) = b \overline{b} w h e r e

z_{0} = \frac{z + z^{'}}{2}

5 . D e d u c e t h a t f o r M \notin (△), M i s

a p e r p e n d i c u l a r b i s e c t o r o f [M M^{'}]