Question

The closest distance of the origin from a curve given as $a\overline{z}+\overline{a}z+a\overline{a}=0$ ($a$ is a complex number) is.

• A. $1$
• B. $\frac{Re\left(|a|\right)}{|a|}$
• C. $\frac{Im\left(|a|\right)}{|a|}$
• D. $\frac{|a|}{2}$

Answer 1

The correct option is D $\frac{|a|}{2}$

The closest distance $=$ length of perpendicular from the origin on the line $a\overline{z}+\overline{a}z+a\overline{a}=0$

$\begin{array}{c}=\frac{a\left(0\right)+\overline{a}\left|0\right|+a\overline{a}}{2\left|a\right|}\\ =\frac{{\left|a\right|}^2}{2\left|a\right|}\\ =\frac{\left|a\right|}{2}\end{array}$

Hence, Option $D$ is the correct answer.