Assertion :If $z_{1} + z_{2} = a$ and $z_{1} z_{2} = b$ where $a = ¯ ¯ ¯ a$ and $b = ¯ ¯ b$ , then $a r g (z_{1} z_{2}) = 0.$ Reason: The sum and product of two complex numbers are real if and only if they are conjugate of each other.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect and Reason is correct

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
Let $z_{1} = a + i b & z_{2} = c + i d$
$z_{1} + z_{2} = a + i b + c + i d = a + c + i (b + d)$
Since $z_{1} + z_{2}$ is real, therefore $b + d = 0$
$\Rightarrow b = - d$
$z_{1} z_{2} = (a + i b) (c + i d) = a c - b d + i (b c + a d)$
Since $z_{1} z_{2}$ is real, therefore $b c + a d = 0$
$∵ b = - d ∴ c = a$
$\Rightarrow z_{2} = a - i b ∴ z_{1} =_{2}$
Since, $z_{1} z_{2}$ is real
$∴ a r g (z_{1} z_{2}) = 0$
Ans: A

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Assertion :If z1+z2=a and z1z2=b where a=¯¯¯a and b=¯¯b, then arg(z1z2)=0. Reason: The sum and product of two complex numbers are real if and only if they are conjugate of each other.

Assertion :If $z_{1} + z_{2} = a$ and $z_{1} z_{2} = b$ where $a = ¯ ¯ ¯ a$ and $b = ¯ ¯ b$ , then $a r g (z_{1} z_{2}) = 0.$ Reason: The sum and product of two complex numbers are real if and only if they are conjugate of each other.