Question

Assertion :If $z_1$ & $z_2$ be two complex numbers such that $|z_1|=|z_2|+|z_1-z_2|$, then Im $\left(\frac{z_1}{z_2}\right)=0$. Reason: arg (z) = 0 $\Rightarrow z$ lies on x - axis.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true (R) is false.
• D. (A) is false (R) is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).

Given arg (z) = 0
$\Rightarrow$ z is purely real $\Rightarrow$ z lies on x-axis
$\Rightarrow$ Reason (R) is true
Again $|z_1|=|z_2|+|z_1+z_2|$
$\therefore (|z_1-z_1|{)}^2=(|z_1|-|z_2|{)}^2$
$\Rightarrow |z_1{|}^2+|z_2{|}^2-2|z_1||z_2|cos({\alpha }_1-{\alpha }_2)=|z_1{|}^2+|z_2{|}^2-2|z_1||z_2|$
$\Rightarrow 2|z_1||z_2|cos({\alpha }_1-{\alpha }_2)=2|z_1||z_2|$
$\Rightarrow cos({\alpha }_1-{\alpha }_2)=1$ $\Rightarrow {\alpha }_1-{\alpha }_2=0$
$\Rightarrow arg\left(z_1\right)-arg\left(z_2\right)=0$
$\Rightarrow arg\left(\frac{z_1}{z_2}\right)=0$
$\Rightarrow z_1/z_2$ is purely real
$\Rightarrow Im\left(\frac{z_1}{z_2}\right)=0$
Assertion (R) is true & Reason (R) is true & Reason (R) is correct explanation of Assertion (A).