Question

If $z_1\ne 0$ and $z_2$ be two complex numbers such that $\frac{z_2}{z_1}$ is a purely imaginary number, then the value of $\left|\frac{2z_1+3z_2}{2z_1-3z_2}\right|$ is

• A. $1$
• B. $3$
• C. $5$
• D. $2$

Answer 1

The correct option is A $1$

Let $\frac{z_2}{z_1}=i\lambda$
Where $\lambda$ is a constant.
Now,
$\left|\frac{2z_1+3z_2}{2z_1-3z_2}\right|=\left|\frac{2z_1+3i\lambda z_1}{2z_1-3i\lambda z_1}\right|=\left|\frac{2+3i\lambda }{2-3i\lambda }\right|(\because z_1\ne 0)=\frac{\sqrt{4+9{\lambda }^2}}{\sqrt{4+9{\lambda }^2}}=1$