Question

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

Short answer

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

Step-by-step solution

Given Information

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

Definition of the Complex number

Every complex number can be represented as:

z=a+bi

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

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 $\left(x,y\right)=\left(\frac{1}{2},0\right)$.