z≠0 and z is complex; z=x+iy with x;y ∈ R^∗ . Given these points with theirs affix: 0(0+0i); N(z^2 −1) and P((1/z^2 )−1) 1. Show that ((1/z^2 )−1)z^2 −1^(−) =−z^2 ^(−) ∣(1/2)−1∣^2 . 2. What is the nature of the set of points M(z) such that O, N and P are aligned?


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Question Number 126267 by mathocean1 last updated on 19/Dec/20

$z \neq 0 a n d z i s c o m p l e x;$ $z = x + i y w i t h x; y \in R^{*} .$ $G i v e n t h e s e p o i n t s w i t h t h e i r s$ $a f f i x :$ $0 (0 + 0 i); N (z^{2} - 1) a n d P (\frac{1}{z^{2}} - 1)$ $1 . S h o w t h a t$ $(\frac{1}{z^{2}} - 1) \overset{―}{z^{2} - 1} = - \overset{―}{z^{2}} ∣ \frac{1}{2} - 1 ∣^{2} .$ $2 . W h a t i s t h e n a t u r e o f t h e s e t o f$ $p o i n t s M (z) s u c h t h a t O, N a n d$ $P a r e a l i g n e d ?$
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