Q55P - Mathematical Methods in Physical Sciences

Question

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

 $\mathit{z}\mathbf{-}\overline{\mathbf{z}}\mathbf{=}\mathbf{5}\mathit{i}$. 

Step-by-Step Solution

Answer

The equation is the equation of the line $y=2.5$ parallel to the x-axis.

Step 1: Given Information

The equation is $z-\overline{z}=5i$.

Step 2: Definition of the Complex number

A complex number can be represented 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 $z-\overline{z}=5i$.

 

The complex number is $z=x+yi$.

 

$\begin{array}{l}z-\overline{z}=x+yi-(x-yi)\\ 5i=2yi\\ y=2.5\end{array}$ 

 

Hence, the equation is the equation of the line $y=2.5$ parallel to the x-axis.