Q60P - Mathematical Methods in Physical Sciences

Question

Describe geometrically the set of points in the complex plane satisfying the following equations.  $\left|z-1+i\right|\mathbf{=}\mathbf{2}$.

Step-by-Step Solution

Answer

This is the equation of the circle with center $\left(1,-1\right)$ and radius $r=2$.

Step 1: Given Information

The equation is $\left|z-1+i\right|=2$.

Step 2: Definition of the Complex number

A complex number can be expressed as:

$\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$
  

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

 

The modulus of a complex number can be calculated as:

 

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

Step 3: Find the value

The equation is $\left|z-1+i\right|=2$.

 

The complex number is $\left|x+yi-1+i\right|=2$.

 

$\begin{array}{l}\left|(x+i)+i(y+1)\right|=2\\ {(x-1)}^2+{(y+1)}^2=4\end{array}$ 

 

Hence, this is the equation of the circle with center $\left(1,-1\right)$ and radius $r=2$.