Question

The complex numbers $z=x+iy$ which satisfy the equation $\left|\frac{\left(z-5i\right)}{\left(z+5i\right)}\right|=1$ lie on which of the following.

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

Answer 1

The correct option is A

The x - axis

Explanation for correct answer :

Finding where z lies:

Given, $z=x+iy$ is the satisfying the equation $\left|\frac{\left(z-5i\right)}{\left(z+5i\right)}\right|=1$.

So, here we are substituting the value of $z$ in equation.

$$\begin{gathered} \Rightarrow \left|\frac{\left(x+iy\right)-5i}{\left(x+iy\right)+5i}\right|=1 \\ \Rightarrow \left|\frac{x+i(y-5)}{x+i(y+5)}\right|=1 \\ \Rightarrow \frac{\sqrt[]{x^2+{\left(y-5\right)}^2}}{\sqrt{x^2+{(y+5)}^2}}=1(\because \left|x+iy\right|=\sqrt{x^2+y^2}) \\ \Rightarrow \sqrt[]{x^2+{\left(y-5\right)}^2}=\sqrt{x^2+{(y+5)}^2} \end{gathered}$$

Squaring both sides we get,

$$\begin{gathered} \Rightarrow x^2+y^2+25-10y=x^2+y^2+25+10y \\ \Rightarrow 20y=0 \\ \Rightarrow y=0 \end{gathered}$$

Hence, $z$ lies on x - axis.

Thus, the correct answer is Option (A).