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

The correct option is B Real axis

We have, $\left|\frac{z-6i}{z+6i}\right|=1$

or, $|z-6i|=|z+6i|$

Let $z=x+iy$

or, $|x+iy-6i|=|x+iy+6i|$

or, $|x+(y-6)i|=|x+(y+6)i|$

or, $\sqrt{x^2+(y-6{)}^2}=\sqrt{x^2+(y+6{)}^2}$

or, $x^2+(y-6{)}^2=x^2+(y+6{)}^2$

or, $y^2+36-12y=y^2+36+12y$

or, $24y=0$

or, $y=0$

Hence, B is the correct option.