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Assertion :If z_1 + z_2 = a and z_1z_2 = b where a = {a} and b

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Assertion :If $z_{1} + z_{2} = a$ and $z_{1} z_{2} = b$ where $a = \overline{a}$ and $b = \overline{b}$ , 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

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

Assertion is correct but Reason is incorrect

Assertion is incorrect and Reason is correct

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William Discussed

Assertion :If

z_{1} + z_{2} = a

and

z_{1} z_{2} = b

where

a = \overline{a}

and

b = \overline{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.

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Text solutionVerified

Let

z_{1} = a + ib & z_{2} = c + i d

z_{1} + z_{2} = a + ib + 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 + ib) (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 - ib ∴ z_{1} = \overset{z_{2}}{ˉ}

Since,

z_{1} z_{2}

is real

∴ a r g (z_{1} z_{2}) = 0

Ans: A

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Question Text	Assertion :If $z_{1} + z_{2} = a$ and $z_{1} z_{2} = b$ where $a = \overline{a}$ and $b = \overline{b}$ , 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.
Answer Type	Text solution:1
Upvotes	150

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

Text solutionVerified

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Assertion :If $z_{1} + z_{2} = a$ and $z_{1} z_{2} = b$ where $a = \overline{a}$ and $b = \overline{b}$ , 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.