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Complex Numbers

If z 1 and ...

Question

If $z_{1}$ and $z_{2}$ are two complex number such that Im $(z_{1} = z_{2})$ = 0= Im $(z_{1} z_{2})$ , then

A

z_{1} = z_{2}

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B

z_{1} = {¯ ¯ ¯ z}_{2}

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C

z_{1} = {- z}_{2}

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D

z_{1} = - {¯ ¯ ¯ z}_{2}

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Solution

The correct option is B $z_{1} = {¯ ¯ ¯ z}_{2}$
As per given condition:

$I m g (z_{1} = z_{2}) = 0 = I m (z_{1} z_{2})$

$\Rightarrow I m (z_{1} z_{2}) = 0$

Product of imaginary parts is zero, implies that the direction of each number is opposite in direction.

$∴ z_{1} =_{2}$

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Q. Assertion :If

z_{1}

z_{2}

be two complex numbers such that

| z_{1} | = | z_{2} | + | z_{1} - z_{2} |

(\frac{z_{1}}{z_{2}}) = 0

. Reason: arg (z) = 0

\Rightarrow z

lies on x - axis.

z_{1}

z_{2}

are two complex numbers, then prove that

| z_{1} | + | z_{2} | = ∣ ∣ ∣ \frac{z_{1} + z_{2}}{2} + \sqrt{z_{1} z_{2}} ∣ ∣ ∣ + ∣ ∣ ∣ \frac{z_{1} + z_{2}}{2} - \sqrt{z_{1} z_{2}} ∣ ∣ ∣

z_{1}, z_{2}

are two complex numbers such that

| z_{1} | = | z_{2} | = 2

arg z_{1} + arg z_{2} = 0

z_{1} z_{2} =

z_{1}, z_{2}

are two complex numbers such that

| z_{1} | = | z_{2} | = 2

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z_{1} z_{2} =

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z_{2}

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| z_{1} | = | z_{2} |

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