Question

If $ \frac{5{{z}_{2}}}{11{{z}_{1}}} $ is purely imaginary, then the value of $ \left[ \frac{2{{z}_{1}}+3{{z}_{2}}}{2{{z}_{1}}-3{{z}_{2}}} \right] $ is

• A. $ \frac{37}{33} $
• B. $ 2 $
• C. 1
• D. $ 3 $
• E. $ \frac{33}{37} $

Answer 1

Answer: (C)

Since, $ \frac{5{{z}_{2}}}{11{{z}_{1}}}=iy $ (say) or $ \frac{{{z}_{2}}}{{{z}_{1}}}=\frac{11}{5}iy $
Now, $ \left| \frac{2{{z}_{1}}+3{{z}_{2}}}{2{{z}_{1}}-3{{z}_{2}}} \right|=\frac{\left| 2+3\frac{{{z}_{2}}}{{{z}_{1}}} \right|}{\left| 2-3\frac{{{z}_{2}}}{{{z}_{1}}} \right|} $
$=\frac{\left| 2+\frac{33}{5}iy \right|}{\left| 2-\frac{33}{5}iy \right|}=1 $