Q59P - Mathematical Methods in Physical Sciences

Question

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

 $\left|z+3i\right|\mathbf{=}\mathbf{4}$. 

Step-by-Step Solution

Answer

This is the equation of the circle with center $\left(0,-3\right)$ and radius $r=4$.

Step 1: Given Information

The equation is $\left|z+3i\right|=4$.

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 $\left|z+3i\right|=4$.

 

The complex number is $|z+3i\left|=\right|x+yi+\overline{)3i}$.

 

$\begin{array}{l}4=x+i(3+y)|\\ 4=\sqrt{x^2+{(3+y)}^2}\\ x^2+{(y+3)}^3=16\end{array}$ 

 

Hence, this is the equation of the circle with center $\left(0,-3\right)$ and radius $r=4$.