Question

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

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

Short answer

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

Step-by-step solution

Given Information

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

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:

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

Find the value

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

The complex number is role="math" localid="1658471525071" $|z+3i\left|=\right|x+yi+\overline{)3i}$.

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

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