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Question: For any two complex numbers \({z_1},{z_2}\) we have \(|{z_1} + {z_2}{|^2} = |{z_1}{|^2} + |{z_2}{|^2...

For any two complex numbers z1,z2{z_1},{z_2} we have ∣z1+z2∣2=∣z1∣2+∣z2∣2|{z_1} + {z_2}{|^2} = |{z_1}{|^2} + |{z_2}{|^2} then,
A. Re⁡(z1z2)=0\operatorname{Re} \left( {\dfrac{{{z_1}}}{{{z_2}}}} \right) = 0
B. Im⁡(z1z2)=0\operatorname{Im} \left( {\dfrac{{{z_1}}}{{{z_2}}}} \right) = 0
C. Re⁡(z1z2)=0\operatorname{Re} ({z_1}{z_2}) = 0
D. Im⁡(z1z2)=0\operatorname{Im} ({z_1}{z_2}) = 0

Explanation

Solution

Hint: In order to solve this problem we need to use the formula of (a+b)2=a2+b2+2ab{(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 ∣z1+z2∣2=∣z1∣2+∣z2∣2|{z_1} + {z_2}{|^2} = |{z_1}{|^2} + |{z_2}{|^2}.
We will apply the formula (a+b)2=a2+b2+2ab{(a + b)^2} = {a^2} + {b^2} + 2ab.
Then the equation ∣z1+z2∣2=∣z1∣2+∣z2∣2|{z_1} + {z_2}{|^2} = |{z_1}{|^2} + |{z_2}{|^2} will become:
⇒∣z1+z2∣2=∣z1∣2+∣z2∣2+2∣z1∣∣z2∣=∣z1∣2+∣z2∣2 ⇒2z1z2=0 Or ⇒z1z2=0  \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 z1=0+0i{z_1} = 0 + 0i then,
Re⁡(z1z2)=0\operatorname{Re} ({z_1}{z_2}) = 0, Im⁡(z1z2)=0\operatorname{Im} ({z_1}{z_2}) = 0, Re⁡(z1z2)=0\operatorname{Re} \left( {\dfrac{{{z_1}}}{{{z_2}}}} \right) = 0 and Im⁡(z1z2)=0\operatorname{Im} \left( {\dfrac{{{z_1}}}{{{z_2}}}} \right) = 0.
Case 2 – If z2=0+0i{z_2} = 0 + 0i then,
Re⁡(z1z2)=0\operatorname{Re} ({z_1}{z_2}) = 0, Im⁡(z1z2)=0\operatorname{Im} ({z_1}{z_2}) = 0
Case 3 – If z1=z2=0+0i{z_1} = {z_2} = 0 + 0i then,
Re⁡(z1z2)=0\operatorname{Re} ({z_1}{z_2}) = 0, Im⁡(z1z2)=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=a2+b2+2ab{(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.