Consider-the-transformation-f-of-the-plane-with-all-points-M-wity-affix-z-mapped-to-the-point-M-with-affix-z-such-that-z-3-i-z-1-i-1-3-1-Given-M-0-the-point-z-0-3-4-3-4-i-calcu

Question Number 88479 by Ar Brandon last updated on 10/Apr/20

C o n s i d e r t h e t r a n s f o r m a t i o n f o f t h e p l a n e w i t h a l l p o i n t s

M w i t y a f f i x z m a p p e d t o t h e p o i n t M^{'} w i t h a f f i x z^{'}

s u c h t h a t z^{'} = - (\sqrt{3} + i) z - 1 + i (1 + \sqrt{3})

1) G i v e n M_{0} t h e p o i n t z_{0} = \frac{\sqrt{3}}{4} + \frac{3}{4} i

c a l c u l a t e {A M}_{0} a n d d e d u c e t h e a n g l e i n r a d i a n s

(T a k i n g A a s t h e c e n t e r o f t h e t r a n s f o r m a t i o n)

2) C o n s i d e r t h e p r o g r e s s i o n w i t h p o i n t s {(M_{n})}_{n ⩾ 0} d e f i n e d b y

f (M_{n}) = M_{n + 1}

a \cdot S h o w b y r e c u r r e n c e t h a t \forall n \in N z_{n} = 2^{n} e^{l n \frac{7 π}{6}} (z_{0} - i)

F i n d {A M}_{n} t h e n d e t e r m i n e t h e s m a l l e s t n a t u r a l n u m b e r, n, s u c h t h a t

{A M}_{n} ⩾ 10^{2}