The complex number(s) $z$ satisfying $z^{n} = (z - 1)^{n}$ in the Argand plane

are concylic

No worries! We‘ve got your back. Try BYJU‘S free classes today!

lie on a line parallel to x axis

No worries! We‘ve got your back. Try BYJU‘S free classes today!

lie on a line parallel to y-axis

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

lie on the vertices of regular polygon of

(n - 1)

sides

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C lie on a line parallel to y-axis
$z^{n} = (z - 1)^{n}$
$\Rightarrow {(\frac{z}{z - 1})}^{n} = 1$
Hence,
$| z | = | z - 1 |$
Let $z = x + i y$
$\Rightarrow x^{2} + y^{2} = x^{2} - 2 x + 1 + y^{2}$
$x = \frac{1}{2}$
Hence, option C is correct.

Suggest Corrections

Similar questions

Q. Prove that all the complex number z which satisfy the equation z^n = (1 + z)^n (n>1) lie on a line parallel to imaginary axis.

Q. The roots of the equation

z^{n} = (z + 1)^{n}

on the complex plane lie on the line

Q. The locus of the complex number

z

satisfying

| z - 1 | = | z - i |

on the argand plane, is

A complex number z is said to be unimodular, if $| z | = 1$ . If and $z_{1}$ and $z_{2}$ are complex numbers such that $\frac{z_{1} - 2 z_{2}}{2 - (z_{1}_{2})}$ is unimodular and $z_{2}$ is not unimodular.
Then, the point $z_{1}$ lies on a

Q. In Argand diagram all the complex numbers z satisfying

| z - 4 i | + | z + 4 i | = 10

lie on a :

Join BYJU'S Learning Program

The complex number(s) z satisfying zn=(z−1)n in the Argand plane

The correct option is C lie on a line parallel to y-axiszn=(z−1)n⇒(zz−1)n=1Hence, |z|=|z−1|Let z=x+iy⇒x2+y2=x2−2x+1+y2x=12Hence, option C is correct.

The complex number(s) $z$ satisfying $z^{n} = (z - 1)^{n}$ in the Argand plane

The correct option is C lie on a line parallel to y-axis
$z^{n} = (z - 1)^{n}$
$\Rightarrow {(\frac{z}{z - 1})}^{n} = 1$
Hence,
$| z | = | z - 1 |$
Let $z = x + i y$
$\Rightarrow x^{2} + y^{2} = x^{2} - 2 x + 1 + y^{2}$
$x = \frac{1}{2}$
Hence, option C is correct.