Complex Numbers 3 Question 6
6. If $z _1$ and $z _2$ are two non-zero complex numbers such that $\left|z _1+z _2\right|=\left|z _1\right|+\left|z _2\right|$, then $\arg \left(z _1\right)-\arg \left(z _2\right)$ is equal to
(1987, 2M)
(a) $-\pi$
(b) $-\frac{\pi}{2}$
(c) 0
(d) $\frac{\pi}{2}$
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Answer:
Correct Answer: 6. (c)
Solution:
- Given, $\left|z _1+z _2\right|=\left|z _1\right|+\left|z _2\right|$
On squaring both sides, we get
$$ \begin{array}{rc} & \left|z _1\right|^{2}+\left|z _2\right|^{2}+2\left|z _1\right|\left|z _2\right| \cos \left(\arg z _1-\arg z _2\right) \\ & =\left|z _1\right|^{2}+\left|z _2\right|^{2}+2\left|z _1\right|\left|z _2\right| \\ \Rightarrow & 2\left|z _1\right|\left|z _2\right| \cos \left(\arg z _1-\arg z _2\right)=2\left|z _1\right|\left|z _2\right| \\ \Rightarrow & \cos \left(\arg z _1-\arg z _2\right)=1 \\ \Rightarrow & \arg \left(z _1\right)-\arg \left(z _2\right)=0 \end{array} $$
