Question

For any two complex numbers ${z_1},{z_2}$ we have $|{z_1} + {z_2}{|^2} = |{z_1}{|^2} + |{z_2}{|^2}$ then,
A. $\operatorname{Re} \left( {\dfrac{{{z_1}}}{{{z_2}}}} \right) = 0$
B. $\operatorname{Im} \left( {\dfrac{{{z_1}}}{{{z_2}}}} \right) = 0$
C. $\operatorname{Re} ({z_1}{z_2}) = 0$
D. $\operatorname{Im} ({z_1}{z_2}) = 0$

Answer 1

Hint: In order to solve this problem we need to use the formula of ${(a + b)^2} = {a^2} + {b^2} + 2ab$ in the given equation and then make cases and solve to get the answer.

Complete step-by-step answer:
The given equation is $|{z_1} + {z_2}{|^2} = |{z_1}{|^2} + |{z_2}{|^2}$.
We will apply the formula ${(a + b)^2} = {a^2} + {b^2} + 2ab$.
Then the equation $|{z_1} + {z_2}{|^2} = |{z_1}{|^2} + |{z_2}{|^2}$ will become:
$
   \Rightarrow |{z_1} + {z_2}{|^2} = |{z_1}{|^2} + |{z_2}{|^2} + 2|{z_1}||{z_2}| = |{z_1}{|^2} + |{z_2}{|^2} \\
   \Rightarrow 2{z_1}{z_2} = 0 \\
  {\text{Or}} \\
   \Rightarrow {z_1}{z_2} = 0 \\
$
For the above any one of the complex number is zero or both of them would be zero (i.e. 0 + 0i)
Case 1 – If ${z_1} = 0 + 0i$ then,
$\operatorname{Re} ({z_1}{z_2}) = 0$, $\operatorname{Im} ({z_1}{z_2}) = 0$, $\operatorname{Re} \left( {\dfrac{{{z_1}}}{{{z_2}}}} \right) = 0$ and $\operatorname{Im} \left( {\dfrac{{{z_1}}}{{{z_2}}}} \right) = 0$.
Case 2 – If ${z_2} = 0 + 0i$ then,
$\operatorname{Re} ({z_1}{z_2}) = 0$, $\operatorname{Im} ({z_1}{z_2}) = 0$
Case 3 – If ${z_1} = {z_2} = 0 + 0i$ then,
$\operatorname{Re} ({z_1}{z_2}) = 0$, $\operatorname{Im} ({z_1}{z_2}) = 0$
Therefore if we consider all the cases since any of it is not mentioned in the question
So, the options A,B,C,D all are answers.

Note: In this problem we have to use the formula we have to use the formula of ${(a + b)^2} = {a^2} + {b^2} + 2ab$ and then we have to make cases. We should have knowledge about what is the real part and imaginary part of a complex number. Since any other information is not provided in the problem. So we need to consider all the conditions. Doing like this you will get the right answer.