Question

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

• A. circle
• B. ellipse
• C. straight line
• D. none of these

Answer 1

The correct option is A circle

$|z-3|=|z-5|$

$\Rightarrow |x+iy-3|=|x+iy-5|$

$\Rightarrow |(x-3)+iy|=|(x-5)+iy|$

Squaring both sides, we get

${(x-3)}^2-y^2+2iy(x-3)={(x-5)}^2-y^2+2iy(x-5)$

$\Rightarrow {(x-3)}^2-{(x-5)}^2=2iy(x-5-x+3)$

$\Rightarrow (x-3-x+5)(x-3+x-5)=-4iy$

$\Rightarrow 2(2x-8)=-4iy$

$\Rightarrow 4(x-4)=-4iy$

$\Rightarrow (x-4)=-iy$

Squaring both sides,we get

${(x-4)}^2={(-iy)}^2$

$x^2-8x+16=-y^2$

or $x^2+y^2-8x+16=0$ is the equation of the circle.