96.7k views
0 votes
Find a number z that satisfies the given equation.
|z| - z = 2 + i

User Janelly
by
8.1k points

1 Answer

7 votes

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

User Fischbrot
by
8.7k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories