Q51P - Mathematical Methods in Physical Sciences

Question

Describe geometrically the set of points in the complex plane satisfying the following equations.  $\left|z\right|\mathbf{=}\mathbf{2}$. 

Step-by-Step Solution

Answer

This is the equation of the circle with a center $\left(0,0\right)$ and radius $r=2$ .

 

Step 1: Given Information

The equation is $\left|Z\right|=2$ .

Step 2: Definition of the Complex number

 A complex number can be written as:

z=x+iy
   

Where z is the complex number, x and y are real numbers, i is known as iota whose value is$\sqrt{-1}$ .

 

The modulus of a complex number can be calculated as:

 

$\left|Z\right|=\sqrt{x^2+y^2}$ 

Step 3: Find the value

The equation is $\left|Z\right|=2$ .

 

So the equation can be figured out as:

 $$\begin{gathered} \left|Z\right|=\left|x+iy\right| \\ 2=\sqrt{x^2+y^2} \\ {x}^2+y^2=4 \end{gathered}$$

 

 Hence, this is the equation of the circle with center $\left(0,0\right)$  and radius r = 2 .