Question

Show thatis the distance between the points and in the complex plane. Use this result to identify the graphs in Problems $\mathbf{53}\mathbf{,}\mathbf{54}\mathbf{,}\mathbf{59}\mathbf{,}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathbf{60}$without computation.

Short answer

The equation has been proven.

Step-by-step solution

Given Information

The complex planes are $z_1,z_2$.

Definition of the Complex number.

A complex number can be dictated 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 $\sqrt{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}$ .

The modulus of a complex number can be calculated as:

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

Prove the statement

The complex planes are $z_1,z_2$.

$$\begin{gathered} \left|z_1-z_2\right|=\left|x_1+y_{1}i-x_2-y_{2}i\right| \\ r=\sqrt{{\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2} \\ {\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2=r^2 \end{gathered}$$

Compare the above equation with the given equation in problem , both have the same form but , so it's an equation of a circle.

Compare the above equation with the given equation in problem , both have the same form, but the outer border is not contained, so it's an equation of disk.

Compare the above equation with the given equation in the problem; both have the same form, so it's an equation of a circle.

Compare the above equation with the given equation in problem , and both have the same form, so it's an equation of the circle.

Hence, the equation has been proven.