Q46 / 90 · JEE Main 2012 (26 May Online) · Mathematics
$\left|z_1+z_2\right|^2+\left|z_1-z_2\right|^2$ is equal to
- A$2\left(\left|z_1\right|+\left|z_2\right|\right.$
- Correct answerB$2\left(\left|z_1\right|^2+\left|z_2\right|^2\right)$
- C$\left|z_1\right|\left|z_2\right|$
- D$\left|z_1\right|^2+\left|z_2\right|^2$
Solution
$\left|z_1+z_2\right|^2+\left|z_1-z_2\right|^2$
$$
=\left|z_1\right|^2+\left|z_2\right|^2+2\left|z_1\right|\left|z_2\right|+\left|z_1\right|^2+\left|z_2\right|^2-2\left|z_1\right|\left|z_2\right|
$$
$$
=2\left|z_1\right|^2+2\left|z_2\right|^2=2\left[\left|z_1\right|^2+\left|z_2\right|^2\right]
$$
Hence, the correct answer is option: 'B'.