Question

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

${\mathit{z}}^{\mathbf{2}}\mathbf{=}\mathbf{-}{\overline{\mathbf{z}}}^{\mathbf{2}}\mathbf{.}$.

Short answer

The equation is$y=\pm x$ , the straight lines.

Step-by-step solution

Given Information

The equation is$z^2=-{\overline{\mathrm{z}}}^2$.

Definition of the Complex number

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

The modulus of a complex number can be calculated as:

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

Find the value.

The equation is $z^2=-{\overline{\mathrm{z}}}^2$.

The complex number is given below as:

$$\begin{gathered} z^2={(x+yi)}^2=x^2-y^2+2xyi \\ {z}^2={(x+yi)}^2 \\ =x^2-y^2+2xyi \end{gathered}$$

Evaluate the value:

$$\begin{gathered} x^2-y^2+2xyi=-(x^2-y^2-2xyi) \\ {x}^2=y^2 \\ y=\pm x \end{gathered}$$

Hence, the equation is $y=\pm x$, the straight lines.