Prove that the point represented by the roots of the equation $z^{4} = {(1 + z)}^{4}$ lie on a straight line. Find the complex equation of the straight lies

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Solution

$\begin{matrix} z^{4} = {(z + 1)}^{4} {| z |}^{4} = {| z + 1 |}^{4} | z | = | z + 1 | \end{matrix}$
Hence, $t$ lies on line $x = \frac{- 1}{2}$ .
Hence roots are collinear.

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