Question

Describe geometrically the set of points in the complex plane satisfying the following equations.

$\left|z-1\right|\mathbf{=}\mathbf{1}$.

Short answer

This is the equation of the circle with a center $\left(1,0\right)$and radius $r=1$.

Step-by-step solution

Given Information

The equation is $\left|Z-1\right|=1$.

Definition of the Complex number

A complex number can be written as:

z=x+iy

Where z is the complex number, x and y are real numbers, and i is known as iota, whose value is$\sqrt{-1}$.

The modulus of a complex number can be calculated as:

$\mathbf{\left|}\mathbf{Z}\mathbf{\right|}\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}$

Find the value

The equation is $\left|Z-1\right|=1$.

The complex number is role="math" localid="1658466205357" $z=x+yi$.

role="math" localid="1658466195116" $\left|x+yi-1\right|=1$

Solve further:

$\begin{array}{l}\sqrt{{(x-1)}^{{}^2}+y^2=1}\\ {(x-1)}^2+y^2=1\end{array}$

Hence, this is the equation of circle with a center $\left(1,0\right)$and radius $r=1$.