Question

If $z$ is a complex number, then $(z+5)(\overline{z}+5)$ is equal to

• A. ${|z+5i|}^2$
• B. ${|z-5|}^2$
• C. ${(z+5)}^2$
• D. ${|z+5|}^2$

Answer 1

The correct option is D ${|z+5|}^2$

$(z+5)(\overline{z}+5)=z\cdot \overline{z}+5(z+\overline{z})+25$

Now $z|\overline{z}|$ $=$ $|z{|}^2$

Using this and simplifying the above equation we have,

$={\left|z\right|}^2+2\cdot 5\left|z\right|+5^2={|z+5|}^2$.