Question

If $z$ is a complex number such that $\left|\frac{z-3i}{z+3i}\right|=1$ then $z$ lies in

• A. the real axis
• B. the line $Im\left(z\right)=3$
• C. a cricle
• D. none of these

Answer 1

Answer: (A)

the real axis

$\left|\frac{z-3i}{z+3i}\right|=1\Rightarrow \frac{z-3i}{z+3i}=\cos A+i\sin A\Rightarrow \frac{2z}{-6i}=\frac{\cos A+i\sin A+1}{\cos A+i\sin A-1}\Rightarrow z=3i\left(\frac{1+\cos A+i\sin A}{1-\cos A-i\sin A}\right)=3i\left(\frac{2{\cos }^2\frac{A}{2}+2i\cos \frac{A}{2}\sin \frac{A}{2}}{2{\sin }^2\frac{A}{2}-2i\cos \frac{A}{2}\sin \frac{A}{2}}\right)\Rightarrow z=3i\cot \frac{A}{2}\left(\frac{\cos \frac{A}{2}+i\sin \frac{A}{2}}{-i\cos \frac{A}{2}+\sin \frac{A}{2}}\right)=3i^2\cot \frac{A}{2}\left(\frac{\cos \frac{A}{2}+i\sin \frac{A}{2}}{\cos \frac{A}{2}+i\sin \frac{A}{2}}\right)\Rightarrow z=-3\cot \frac{A}{2}$
Therefore z lies on x axis.
Ans: A