Let z be a complex number satisfying |z| ^ 3 + 2 z ^ 2 + 4 ¯ z - 8...

Given,

${\left|z\right|}^{3} + 2{z}^{2} + 4\overset{¯}{z} - 8 = 0 . . . . . . . \left(1\right)$

Now on taking conjugate both side we get,

${\left|z\right|}^{3} + 2\overset{¯}{z}2 + 4z - 8 = 0 . . . . . . . . \left(2\right)$

\left\{\text{Note}\left|\overset{¯}{z}\right| = \left|z\right|\right\}$</p><p>Now subtracting both equations we get,</p><p>$2\left({z}^{2} - \overset{¯}{z}2\right) + 4\left(\overset{¯}{z} - z\right) = 0$</p><p>$⇒ \left(z - \overset{¯}{z}\right)\left[2\left(z + \overset{¯}{z}\right) - 4\right] = 0$</p><p>$∵ z = \overset{¯}{z}$(not possible) or$4x = 4 ⇒ x = 1$\left\{\text{as}z + \overset{¯}{z} = 2x\right\}$

So, $z = 1 + \lambdai$

$⇒ \left|z\right| = \sqrt{1 + {\lambda}^{2}} \text{ and } \overset{¯}{z} = 1 - \lambdai$

Now putting the value of $z, \left|z\right| \text{ and } \overset{¯}{z}$ in given equation we get,

${\left(1 + {\lambda}^{2}\right)}^{3 / 2} + 2\left(1 - {\lambda}^{2} + 2\lambdai\right) + 4\left(1 - \lambdai\right) - 8 = 0$

$⇒ {\left(1 + {\lambda}^{2}\right)}^{3 / 2} + 2\left(1 - {\lambda}^{2}\right) = 4$

$⇒ {\left(1 + {\lambda}^{2}\right)}^{3 / 2} = 2\left(1 + {\lambda}^{2}\right)$

$⇒ \left(1 + {\lambda}^{2}\right)\left[\sqrt{1 + {\lambda}^{2}} - 2\right] = 0$

$⇒ {\lambda}^{2} = 3$

Now solving,

\left(P\right)${\left|z\right|}^{2} = 1 + {\lambda}^{2} = 1 + 3 = 4$

\left(Q\right)${\left|z - \overset{¯}{z}\right|}^{2} = {\left|1 + \lambdai - \left(1 - \lambdai\right)\right|}^{2} = {\left|2\lambdai\right|}^{2} = 4{\lambda}^{2} = 12$

\left(R\right)${\left|z\right|}^{2} + {\left|z + \overset{¯}{z}\right|}^{2} = 4 + {\left|\left(1 + \lambdai\right) + \left(1 - \lambdai\right)\right|}^{2} = 4 + 4 = 8$

\left(S\right)${\left|z + 1\right|}^{2} = {\left|1 + \lambdai + 1\right|}^{2} = 4 + {\lambda}^{2} = 4 + 3 = 7$

$∴ P \to \left(2\right), Q \to \left(1\right), R \to \left(3\right), S \to \left(5\right)$