First slide

Geometry of complex numbers

Question

All the roots of $(z + 1)^{4} = z^{4}$ lie on

Moderate

A

a straight line parallel to x-axis
B

a straight line parallel to y-axis
C

a circle with centre at $- 1 + 0 i$
D

a circle with centre at $1 + i$

Solution

TIP
It is unnecessary to find roots of $(z + 1)^{4} = z^{4}$
If $z$ is a root of $(z + 1)^{4} = z^{4},$ then $|(z + 1)^{4}| = |z^{4}|$
$\begin{array}{l} \Rightarrow | z + 1 | = | z | \\ \Rightarrow | z - (- 1) | = | z - 0 | \end{array}$
$\Rightarrow z$ lies on the perpendicular bisector of the segment join
$ing - 1 + 0 i and 0 i.e. z$ z lies on the line $Re (z) = - 1 / 2 .$

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