Q50P - Mathematical Methods in Physical Sciences

Question

Solve for all possible values of the real numbers x and y in the following equations. $\mathbf{|}\mathbf{x}\mathbf{+}\mathbf{i}\mathbf{y}\mathbf{|}\mathbf{=}\mathit{y}\mathbf{-}\mathit{i}\mathit{x}\mathbf{.}$ .

Step-by-Step Solution

Answer

The possible values are given below:

 $x=0,y\ge 0$

Step 1: Given Information

The equation is $|x+iy|=y-ix.$  .

Step 2: Definition of the Complex number

Every complex number can be represented as: 

 

 z=a+bi

 

Where a and b are both real numbers, z is the complex number, and i is known as iota, which makes z a complex number.

 

Step 3: Find the values

 The equation is $\left|x+yi\right|=y-xi.$ .

 

The value of the modulus must be real and positive so, $x=0$ .

 

Now the above equation can be written as: 

$y=\left|iy\right|$

 

Hence, it satisfies all the values of any real positive number.

 

The possible values are given below:

$x=0,y\ge 0$