SATHEE: Complex Numbers 3 Question 6

Complex Numbers 3 Question 6

6. If $z_{1}$ and $z_{2}$ are two non-zero complex numbers such that $| z_{1} + z_{2} | = | z_{1} | + | z_{2} |$ , then $\arg (z_{1}) - \arg (z_{2})$ is equal to

(1987, 2M)

(a) $- π$

(b) $- \frac{π}{2}$

(c) 0

(d) $\frac{π}{2}$

Show Answer

Answer:

Correct Answer: 6. (c)

Solution:

Given, $| z_{1} + z_{2} | = | z_{1} | + | z_{2} |$

On squaring both sides, we get

$\begin{array}{rc} {| z_{1} |}^{2} + {| z_{2} |}^{2} + 2 | z_{1} | | z_{2} | \cos (\arg z_{1} - \arg z_{2}) \\ = {| z_{1} |}^{2} + {| z_{2} |}^{2} + 2 | z_{1} | | z_{2} | \\ \Rightarrow & 2 | z_{1} | | z_{2} | \cos (\arg z_{1} - \arg z_{2}) = 2 | z_{1} | | z_{2} | \\ \Rightarrow & \cos (\arg z_{1} - \arg z_{2}) = 1 \\ \Rightarrow & \arg (z_{1}) - \arg (z_{2}) = 0 \end{array}$

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Complex Numbers 3 Question 6

6. If z1 and z2 are two non-zero complex numbers such that |z1+z2|=|z1|+|z2|, then arg⁡(z1)−arg⁡(z2) is equal to

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6. If $z_{1}$ and $z_{2}$ are two non-zero complex numbers such that $| z_{1} + z_{2} | = | z_{1} | + | z_{2} |$ , then $\arg (z_{1}) - \arg (z_{2})$ is equal to