Q49P - Mathematical Methods in Physical Sciences

Question

Solve for all possible values of the real numbers x and y in the following equations. $\left|1-\left(x+iy\right)\right|\mathbf{=}\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$

Step-by-Step Solution

Answer

The only possible value is $\left(x,y\right)=\left(\frac{1}{2},0\right)$.

Step 1: Given Information

The equation is $\left|1-\left(x+yi\right)\right|=x+yi.$

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 $\left|1-\left(x+yi\right)\right|=x+yi.$  .

 

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

 

Now the equation can be written as:

 $$\begin{gathered} x=\left|1-x\right| \\ x=\sqrt{1-x^2} \\ {x}^2=1-x^2 \\ x=\frac{1}{2} \end{gathered}$$

 

 Thus, the only possible value is $\left(x,y\right)=\left(\frac{1}{2},0\right)$ .