Question

The points representing complex number $2$ for which $ \left| z-3 \right|=\left| z-5 \right| $ lie on the locus given by

• A. an ellipse
• B. a circle
• C. a straight line
• D. None of the above

Answer 1

Answer: (C)

Given, $|z-3|=|z-5|$
On squaring both sides, we get
$(z-3)(\bar{z}-3)=(z-5)(\bar{z}-5)$
$\Rightarrow z \bar{z}-3 \bar{z}-3 z+9=z \bar{z}-5 \bar{z}-5 z+25$
$\Rightarrow 2 \bar{z}+2 z=16$
$\Rightarrow z+\bar{z}=8$
$\Rightarrow 2 x=8$
$\Rightarrow x=4$ (putting $z=x+i y)$
Hence, locus of $z$ is a straight line parallel to $y$ -axis.