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Mathematics Question on Complex Numbers and Quadratic Equations

If 5z211z1\frac{5{{z}_{2}}}{11{{z}_{1}}} is purely imaginary, then the value of [2z1+3z22z1−3z2]\left[ \frac{2{{z}_{1}}+3{{z}_{2}}}{2{{z}_{1}}-3{{z}_{2}}} \right] is

A

3733\frac{37}{33}

B

22

C

1

D

33

Answer

1

Explanation

Solution

Since, 5z211z1=iy\frac{5{{z}_{2}}}{11{{z}_{1}}}=iy (say) or z2z1=115iy\frac{{{z}_{2}}}{{{z}_{1}}}=\frac{11}{5}iy
Now, ∣2z1+3z22z1−3z2∣=∣2+3z2z1∣∣2−3z2z1∣\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|}
=∣2+335iy∣∣2−335iy∣=1=\frac{\left| 2+\frac{33}{5}iy \right|}{\left| 2-\frac{33}{5}iy \right|}=1