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