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 calculate AM_0 and deduce the angle in radians (Taking A as the center of the transformation) 2) Consider the progression with points(M_n )_(n≥0) defined by f(M_n )=M_(n+1) a∙ Show by recurrence that ∀n∈N z_n =2^n e^(ln((7π)/6)) (z_(0 ) −i) Find AM_n then determine the smallest natural number, n, such that AM


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

$C o n s i d e r t h e t r a n s f o r m a t i o n f o f t h e p l a n e w i t h a l l p o i n t s$ $M w i t y a f f i x z m a p p e d t o t h e p o i n t M^{'} w i t h a f f i x z^{'}$ $s u c h t h a t z^{'} = - (\sqrt{3} + i) z - 1 + i (1 + \sqrt{3})$ $1) G i v e n M_{0} t h e p o i n t z_{0} = \frac{\sqrt{3}}{4} + \frac{3}{4} i$ $c a l c u l a t e {A M}_{0} a n d d e d u c e t h e a n g l e i n r a d i a n s$ $(T a k i n g A a s t h e c e n t e r o f t h e t r a n s f o r m a t i o n)$ $2) C o n s i d e r t h e p r o g r e s s i o n w i t h p o i n t s {(M_{n})}_{n ⩾ 0} d e f i n e d b y$ $f (M_{n}) = M_{n + 1}$ $a \cdot S h o w b y r e c u r r e n c e t h a t \forall n \in N z_{n} = 2^{n} e^{l n \frac{7 π}{6}} (z_{0} - i)$ $F i n d {A M}_{n} t h e n d e t e r m i n e t h e s m a l l e s t n a t u r a l n u m b e r, n, s u c h t h a t$ ${A M}_{n} ⩾ 10^{2}$
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