Geometry of complex numbers

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$If 1 + 2 {|z|}^{2} = {|z^{2} + 1|}^{2} + 2 {|z^{2} + 1|}^{2},then the calue of |z (z + 1)| is$

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Solution

$1 + 2 {|z|}^{2} = {|z^{2} + 1|}^{2} + 2 {|z^{2} + 1|}^{2}$
$\Rightarrow 1 + 2 z \bar{z} = (z^{2} + 1) ({\bar{z}}^{2} + 1) + 2 (z + 1) (\bar{z} + 1)$
$\begin{array}{l} \Rightarrow (z \bar{z})^{2} + 2 (z + \bar{z}) + z^{2} + {\bar{z}}^{2} + 2 = 0 \\ \Rightarrow (z \bar{z} - 1)^{2} + (z + \bar{z} + 1)^{2} = 0 \\ \Rightarrow z \bar{z} = 1 \\ \Rightarrow \bar{z} = \frac{1}{z} \end{array}$
$\begin{array}{ll} and z + \bar{z} + 1 = 0 \\ or z + \frac{1}{z} + 1 = 0 \\ \Rightarrow z^{2} + z = - 1 \\ \Rightarrow |z^{2} + z| = 1 \end{array}$

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Geometry of complex numbers

Remember concepts with our Masterclasses.

If 1+2z2=z2+12+2z2+12,then the calue of zz+1is

1+2z2=z2+12+2z2+12⇒1+2zz¯=z2+1z¯2+1+2z+1z¯+1⇒ (zz¯)2+2(z+z¯)+z2+z¯2+2=0⇒ (zz¯−1)2+(z+z¯+1)2=0⇒ zz¯=1⇒ z¯=1z and z+z¯+1=0 or z+1z+1=0⇒ z2+z=−1⇒ z2+z=1

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$If 1 + 2 {|z|}^{2} = {|z^{2} + 1|}^{2} + 2 {|z^{2} + 1|}^{2},then the calue of |z (z + 1)| is$