InterviewSolution
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InterviewSolution
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Let ` z_1 and z_2` be complex numbers of such that `z_1!=z_2 and |z_1|=|z_2|. If z_1` has positive real part and `z_2` has negative imginary part, then which of the following statemernts are correct for te vaue of `(z_1+z_2)/(z_1-z_2)` (A) 0 (B) real and positive (C) real and negative (D) purely imaginaryA. zeroB. real and positiveC. real and negativeD. purely imaginary |
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Answer» Correct Answer - A::D Given `|z_1|=|z_2|` Now `(z_1+z_2)/(z_1-z_2)xx(barz_1-barz_2)/(z_1-z_2)=(z_1barz_1-z_1barz_2+z_2barz_1-z_2barz_2)/(|z_1-z_2|)` `=(|z|^2+(z_2barz_1-z_1barz_2)-|z_2|^2)/(|z_1-z_2|^2)` `=(z_2barz_1-z_1barz_2)/(|z_1-z_2|^2)" "[because |z_1|^2=|z_2|^2]` As we know `z-bar z `=2 i Im (z) `therefore ( z_2barz_1-z_1barz_2=2i Im (z_2 bar z_1)` `therefore (z_1+z_2)/(z_1-z_2)=(2i Im(z_2 bar z_1))/(|z_1-z_2|^2)` Which is purely imaginary or zero. |
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