Question

The complex number Z=x+iy which satisfy the equation mod(Z-5i / Z+5i )=1 , lie on (a)The x- axis (b)The straight line y=5 (c)A circle passing through through the origin (d)None.

Answer 1

$$\begin{gathered} \left|\frac{\mathrm{z}-5\mathrm{i}}{\mathrm{z}+5\mathrm{i}}\right|=1 \\ \frac{\left|\mathrm{x}+\mathrm{i}\left(\mathrm{y}-5\right)\right|}{\left|\mathrm{x}+\mathrm{i}\left(\mathrm{y}+5\right)\right|}=1 \\ \sqrt{{\mathrm{x}}^2+{\left(\mathrm{y}+5\right)}^2}=\sqrt{{\mathrm{x}}^2+{\left(\mathrm{y}-5\right)}^2} \\ \mathrm{Squaring}\mathrm{we}\mathrm{get} \\ {\left(\mathrm{y}+5\right)}^2={\left(\mathrm{y}-5\right)}^2 \\ 20\mathrm{y}=0 \\ \mathrm{y}=0 \\ \mathrm{So}\mathrm{it}\mathrm{is}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{x}-\mathrm{axis}. \end{gathered}$$