If a complex number z satisfies |z|$$2+4$$|z|$$2$$−$$2zz$$¯$$+z$$¯z−$$16=0$$, then the maximum value of Zis

Question

If a complex number z satisfies |z|$$2+4$$|z|$$2$$−$$2zz$$¯$$+z$$¯z−$$16=0$$, then the maximum value of Zis

Infinity Learn · Accepted Answer

Given that $$z2+4$$|z|$$2$$−$$2zz$$¯$$+z$$¯z−$$16=0$$……  (1) Put $$z=r($$$$cos$$⁡θ$$+isin$$⁡θ$$)$$⇒z¯$$=r($$$$cos$$⁡θ−isin⁡θ$$)$$ And zz¯$$=$$$$cos$$⁡$$2$$θ$$+isin$$⁡$$2$$θ,z¯$$z=$$$$cos$$⁡$$2$$θ−isin⁡$$2$$θ|z¯|$$2=r2$$ Substitute all these in equation (1) ,we get $$r2+4r2$$−$$4cos$$⁡$$2$$θ−$$16=0$$⇒$$r4+4$$−$$r2(4cos$$⁡$$2$$θ$$+16)=0$$ Put $$r2=t$$⇒$$t2$$−$$t(4cos$$⁡$$2$$θ$$+16)+4=0$$⇒$$t=4cos$$⁡$$2$$θ$$+16$$±(4cos$$⁡$$2$$θ$$+16)2$$−$$162$$⇒$$r2=2($$$$cos$$⁡$$2$$θ$$+4)±$$2($$$$cos$$⁡$$2$$θ$$+4)2$$−$$1$$∴ Maximum of $$r2=10+224$$  , when $$cos2$$θ$$=1=6+4+26$$×$$4=(6+4)2$$∴ Maximum of $$=2+6$$

$If a complex number z satisfies | z |^{2} + \frac{4}{| z |^{2}} - 2 (\frac{z}{\bar{z}} + \frac{\bar{z}}{z}) - 16 = 0, then the maximum value of |Z| is$

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If a complex number z satisfies |z|2+4|z|2−2zz¯+z¯z−16=0, then the maximum value of Zis

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$If a complex number z satisfies | z |^{2} + \frac{4}{| z |^{2}} - 2 (\frac{z}{\bar{z}} + \frac{\bar{z}}{z}) - 16 = 0, then the maximum value of |Z| is$