Question

The complex number $z=x+iy$ which satisfies the equation $\left|\frac{z-3i}{z+3i}\right|=1$lies on

• A. the X-axis
• B. the straight line $y=3$
• C. a circle passing through origin
• D. None of the above

Answer 1

The correct option is A the X-axis

$\left|\frac{z-3i}{z+3i}\right|=1$
$\Rightarrow \frac{|z-3i|}{|z+3i|}=1$
$\Rightarrow |z-3i|=|z+3i|$
$\Rightarrow |x+iy-3i|=|x+iy+3i|$
$squaringbothsides$
${|x+iy-3i|}^2={|x+iy+3i|}^2$
$\Rightarrow {|x+i(y-3)|}^2={|x+i(y+3)|}^2$
$\Rightarrow x^2+{(y-3)}^2=x^2+{(y+3)}^2$
$\Rightarrow x^2+y^2+9-6y=x^2+y^2+9+6y$
$\Rightarrow 12y=0$
$\Rightarrow y=0$
$x-axis$