Question

The complex number $z=x+iy$ which satisfies the equation $\left|\frac{\left(z-3i\right)}{\left(z+3i\right)}\right|=1$ lie on which of the following.

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

Answer 1

The correct option is A

The $X$- axis

Explanation of correct answer :

Determining where complex number $z$ lies ,

Given, $z=x+iy$ and $\left|\frac{\left(z-3i\right)}{\left(z+3i\right)}\right|=1$

Substituting the value of $z$ in the given equation, we get

$$\begin{gathered} \Rightarrow \left|\frac{\left(x+iy\right)-3i}{(x+iy)+3i}\right|=1 \\ \Rightarrow \left|\frac{x+i(y-3)}{x+i(y+3)}\right|=1 \\ \Rightarrow \frac{\sqrt[]{x^2+{(y-3)}^2}}{\sqrt{x^2+{(y+3)}^2}}=1\left[\because x+iy=\sqrt{x^2+y^2}\right] \\ \Rightarrow \sqrt{x^2+{(y-3)}^2}=\sqrt{x^2+{(y+3)}^2} \end{gathered}$$

Squaring both the sides we get,

$$\begin{gathered} \Rightarrow x^2+y^2+9-6y=x^2+y^2+9+6y \\ \Rightarrow 12y=0 \\ \Rightarrow y=0 \end{gathered}$$

Since the obtained value of $y=0$, the complex number $z$ lies on the $X$-axis.

Hence, option (A) is the correct answer.