If - Z 1- Z 2- Z 1- Z 2- where Z 1- Z 2 are two non-zero complex numbers- then Z 1- Z 2 isA- -B- - - 2C- 0D- - - 2

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Properties of Argument

If | Z 1+ Z 2...

Question

If | $Z_{1}$ + $Z_{2}$ | = | $Z_{1}$ | - | $Z_{2}$ | where $Z_{1}$ , $Z_{2}$ are two non-zero complex numbers, then arg( $Z_{1}$ ) - arg( $Z_{2}$ ) is

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π/2

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- π/2

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Solution

The correct option is A
π

| $Z_{1}$ + $Z_{2}$ | = | $Z_{1}$ | - | $Z_{2}$ |

Squaring on both sides,

$| Z_{1} + Z_{2} |^{2}$ = $| z_{1} |^{2}$ + $| z_{2} |^{2}$ - 2 $| z_{1} |$ $| z_{2} |$

$| z_{1} |^{2}$ + $| z_{2} |^{2}$ + 2 $| z_{1} |$ $| z_{2} |$ $c o s (a r g (z_{1}) - a r g (z_{2}))$
= $| z_{1} |^{2}$ + $| z_{2} |^{2}$ - 2 $| z_{1} |$ $| z_{2} |$

$\Rightarrow$ $c o s (a r g (z_{1}) - a r g (z_{2}))$ = -1 = cos $Π$

$\Rightarrow$ $a r g (z_{1}) - a r g (z_{2})$ = $Π$

So, A is right option.

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Similar questions

if $Z_{1}$ and $Z_{2}$ are 2 non - complex number such that | $Z_{1}$ + $Z_{2}$ | = | $Z_{1}$ | + | $Z_{2}$ |,Then arg ( $Z_{1}$ ) - arg( $Z_{2}$ ) is equal to

Hint: Square on both sides.

Q. If

z_{1}

and

z_{2}

are two non-zero complex numbers such that

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

then

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Q. If z₁ and z₂ are two non-zero complex such that

∣ z_{1} + z_{2} ∣=∣ z_{1} ∣ + ∣ z_{2} ∣,

then arg

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If $| z_{1} + z_{2} | = | z_{1} | + | z_{2} |$ where $z_{1}$ and $z_{2}$ are different non-zero complex numbers, then :

z_{1}

and

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are two non-zero complex numbers such that

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

and

a r g z_{1} + a r g z_{2} = π

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If |Z1 + Z2| = |Z1| - |Z2| where Z1, Z2 are two non-zero complex numbers, then arg(Z1) - arg(Z2) is

The correct option is A π |Z1 + Z2| = |Z1| - |Z2| Squaring on both sides, |Z1+Z2|2 = |z1|2 + |z2|2 - 2|z1| |z2| |z1|2 + |z2|2 + 2|z1| |z2| cos(arg(z1)−arg(z2)) = |z1|2 + |z2|2 - 2|z1| |z2| ⇒ cos(arg(z1)−arg(z2)) = -1 = cosΠ ⇒ arg(z1) −arg(z2) = Π So, A is right option.

If | $Z_{1}$ + $Z_{2}$ | = | $Z_{1}$ | - | $Z_{2}$ | where $Z_{1}$ , $Z_{2}$ are two non-zero complex numbers, then arg( $Z_{1}$ ) - arg( $Z_{2}$ ) is