Question

All complex numbers z which satisfy the equation $\left|\frac{z-6i}{z+6i}\right|=1$ lie on the

• A. imaginary axis
• B. real axis
• C. neither of the axes
• D. none of these

Answer 1

Answer: (B)

Let $z=x+iy$
$\therefore \left|\frac{x+iy-6i}{x+iy+6i}\right|=1$ or $\frac{|x+i\left(y-6\right)|}{|x+i\left(y+6\right)|}=1$
$\Rightarrow \sqrt{x^2+{\left(y-6\right)}^2}=\sqrt{x^2+{\left(y+6\right)}^2}$
$\Rightarrow x^2+y^2+36-12y=x^2+y^2+36+12y$
$\Rightarrow 24y=0\Rightarrow y=0$
$\therefore z=x$ i.e., $z$ lies on the real axis.