Question

The complex numbers $z=x+iy$ which satisfy the equation $\left|\right(z-5i\left)/\right(z+5i\left)\right|=1$ lie on

• 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

$\left|\frac{z-5i}{z+5i}\right|=1\Rightarrow \frac{z-5i}{z+5i}=\cos A+i\sin A\Rightarrow \frac{2z}{-10i}=\frac{\cos A+i\sin A+1}{\cos A+i\sin A-1}\Rightarrow z=5i\left(\frac{1+\cos A+i\sin A}{1-\cos A-i\sin A}\right)=5i\left(\frac{2{\cos }^2\frac{A}{2}+2i\cos \frac{A}{2}\sin \frac{A}{2}}{2{\sin }^2\frac{A}{2}-2i\cos \frac{A}{2}\sin \frac{A}{2}}\right)\Rightarrow z=5i\cot \frac{A}{2}\left(\frac{\cos \frac{A}{2}+i\sin \frac{A}{2}}{-i\cos \frac{A}{2}+\sin \frac{A}{2}}\right)=5i^2\cot \frac{A}{2}\left(\frac{\cos \frac{A}{2}+i\sin \frac{A}{2}}{\cos \frac{A}{2}+i\sin \frac{A}{2}}\right)\Rightarrow z=-5\cot \frac{A}{2}$
Therefore z lies on x axis.
Ans: A