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Question Number 126267 by mathocean1 last updated on 19/Dec/20
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?
$${z}\neq\mathrm{0}\:{and}\:{z}\:{is}\:{complex}; \\ $$$${z}={x}+{iy}\:{with}\:{x};{y}\:\in\:\mathbb{R}^{\ast} . \\ $$$${Given}\:{these}\:{points}\:{with}\:{theirs} \\ $$$${affix}: \\ $$$$\mathrm{0}\left(\mathrm{0}+\mathrm{0}{i}\right);\:{N}\left({z}^{\mathrm{2}} −\mathrm{1}\right)\:{and}\:{P}\left(\frac{\mathrm{1}}{\mathrm{z}^{\mathrm{2}} }−\mathrm{1}\right) \\ $$$$\mathrm{1}.\:{Show}\:{that} \\ $$$$\left(\frac{\mathrm{1}}{{z}^{\mathrm{2}} }−\mathrm{1}\right)\overline {{z}^{\mathrm{2}} −\mathrm{1}}=−\overline {{z}^{\mathrm{2}} }\mid\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{1}\mid^{\mathrm{2}} . \\ $$$$\mathrm{2}.\:\mathscr{W}{hat}\:{is}\:{the}\:{nature}\:{of}\:{the}\:\:{set}\:{of}\: \\ $$$${points}\:{M}\left({z}\right)\:{such}\:{that}\:{O},\:{N}\:{and} \\ $$$${P}\:\:\:{are}\:{aligned}? \\ $$$$ \\ $$

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