Question

if z₁ and z₂ are complex numbers

5z₂ / 11z₁ is purely imaginary then find the mode of 2z₁+3z₂ / 2z₁-3z₂

Answer 1

Given
5z₂ / 11z₁ is purely imaginary
$$\begin{gathered} Let\frac{5z_2}{11z_1}=iy \\ \frac{z_2}{z_1}=\frac{11iy}{5} \\ Now, \\ \frac{2z_1+3z_2}{2z_1-3z_2}=\frac{z_1(2+3{\displaystyle \frac{z_2}{z_1}})}{z_1(2-3{\displaystyle \frac{z_2}{z_1}})}=\frac{(2+3\left(\frac{11iy}{5}\right))}{(2-3\left(\frac{11iy}{5}\right))}=\frac{10+33iy}{10-33iy} \\ Now,\left|\frac{2z_1+3z_2}{2z_1-3z_2}\right|=\left|\frac{10+33iy}{10-33iy}\right|=\frac{\sqrt{100+1089}}{\sqrt{100+1089}}=1 \end{gathered}$$