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

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

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

ANSWER :

$z=-\frac{3}{4}-i$