Question

Assertion :Let $z_1$ and $z_2$ be two non-zero complex numbers such that $|z_1+z_2|=|z_1-z_2|$ then $Re\left(\frac{z_1}{z_2}\right)=0$ Reason: If diagonals of a parallelogram are equal then it is a square.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Given two non-zero complex number such that

$\Rightarrow |z_1+z_2|=|z_1-z_2|$

Let $z_1=a+ib$ $z_1=c+id$

$\Rightarrow |z_1+z_2|=|z_1-z_2|$

$\Rightarrow {|z_1+z_2|}^2={\left|z_1\right|}^2+2\left|z_1{z}_2\right|+{\left|z_2\right|}^2={(a+ib)}^2+{(c+id)}^2+2\left|z_1{z}_2\right|$

${|z_1-z_2|}^2={\left|z_1\right|}^2-2\left|z_1{z}_2\right|+{\left|z_2\right|}^2={(a+ib)}^2+{(c+id)}^2-2\left|z_1{z}_2\right|$

${|z_1+z_2|}^2={|z_1-z_2|}^2$

$\Rightarrow 2\left|z_1{z}_2\right|=-2\left|z_1{z}_2\right|$

$4\left|z_1{z}_2\right|=0$ $\left|z_1{z}_2\right|=\left|z_1\right|\left|z_2\right|$

$\Rightarrow \left|z_1{z}_2\right|=0$ $\left|z_2\right|\left|{z^{\prime }}_2\right|=\left|{z_1{z}^{\prime }}_2\right|$

i.e, $Re\left({z_1{z}^{\prime }}_2\right)=\left|{z_1{z}^{\prime }}_2\right|$

Them $Im\left(z\right)=0$

$Re\left(\frac{z_1}{z_2}\right)=0$

The reason 'If diagonals of a parallelogram are equal then it is a square is wrong reason for the given assertion.

$\therefore$ Assertion is correct but reason is in correct.

Hence, the answer is Assertion is correct but reason is in correct.