Find a number z that satisfies the given equation. |z| - z = 2 + i

Question

Find a number z that satisfies the given equation.
`|z| - z = 2 + i`

Answer 1

Let z = x + yi

Then :

`|z|=\sqrt[]{x^2+y^2}`

From the problem, we have :

`\begin{gathered} |z|-z=2+i \\ \text{Substitute the given expressions :} \\ \sqrt[]{x^2+y^2}-(x+yi)=2+i \\ \sqrt[]{x^2+y^2}-x-yi=2+i \end{gathered}`

Comparing the imaginary parts :

`\begin{gathered} -yi=i \\ \text{Therefore :} \\ y=-1 \end{gathered}`

Substitute y = -1

`\begin{gathered} \sqrt[]{x^2+y^2}-x-yi=2+i \\ \sqrt[]{x^2+(-1)^2}-x-(-1)i=2+i \\ \sqrt[]{x^2+1}-x+i=2+i \\ \sqrt[]{x^2+1}=2+i+x-i \\ \sqrt[]{x^2+1}=2+x \\ \text{Square both sides :} \\ x^2+1=4+4x+x^2 \\ 1-4=4x+x^2-x^2 \\ -3=4x \\ x=-(3)/(4) \end{gathered}`

The real part is x = -3/4 and the imaginary part is y = -1

ANSWER :

`z=-(3)/(4)-i`