Question

If $z_1$ and $z_2$ are two complex numbers such that $|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2$, then

• A. $z_1\overline{z_2}$ is purely imaginary
• B. $z_1/z_2$ is purely imaginary
• C. $z_1\overline{z_2}+\overline{z_1}{z}_2=0$
• D. $O,z_1,z_2$ are vertices of a right triangle

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $z_1/z_2$ is purely imaginary
B $O,z_1,z_2$ are vertices of a right triangle
C $z_1\overline{z_2}+\overline{z_1}{z}_2=0$
D $z_1\overline{z_2}$ is purely imaginary

$|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2$
$\Rightarrow (z_1+z_2)(\overline{z_1}+\overline{z_2})=|z_1{|}^2+|z_2{|}^2$
$\Rightarrow z_1\overline{z_2}+\overline{z_1}{z}_2=0$
$\Rightarrow z_1\overline{z_2}+\overline{z_1\overline{z_2}}=0$
Therefore, $z_1\overline{z_2}$ is purely imaginary
$\Rightarrow \frac{z_1}{z_2}+\frac{\overline{z_1}}{\overline{z_2}}=0$
also $\frac{z_1}{z_2}$ is purely imaginary.
$\Rightarrow arg\left(\frac{z_1}{z_2}\right)=\frac{\pi }{2}$
and hence, $O,z_1,z_2$ are vertices of right triangle.
Ans: A,B,C,D