Question

The complex number $ z = x + iy $ satisfy the equation $ |\frac {z-5i}{z+5i}| = 1 $ lies on

• A. the x-axis
• B. the straight line $y = 5$
• C. a circle passing through origin
• D. None of the above

Answer 1

Answer: (A)

We have, $z=x+i y$
$\therefore \left|\frac{z-5 i}{z+5 i}\right|=1$
$\Rightarrow |z-5 i|=|z+5 i|$
$\Rightarrow |x+i y-5 i|=|x+i y+5 i|$
$\Rightarrow |x+(y-5) i|=|x+(y+5) i|$
$\Rightarrow \sqrt{x^{2}+(y-5)^{2}}=\sqrt{x^{2}+(y+5)^{2}}$
$\Rightarrow x^{2}+(y-5)^{2}=x^{2}+(y+5)^{2}$
$\Rightarrow (y-5)^{2}=(y+5)^{2}$
$\Rightarrow y^{2}-10 y+25=y^{2}+10 y+25$
$\Rightarrow 20 y=0$
$\Rightarrow y=0$
$\therefore z$ lies on $X$-axis.