If -z1-z2-z1-z2- then arg z1- z2-

The correct option is C |z_1 + z_2|^2 = |z_1 z_2|^2 ⇒ z_1.z_2 + z_1.z_2 =0 i.e, z_1.z_2 + z_1.z_2 =0 ⇒ Re(z_1.z_2) =0 Put z_1=r_1e^i^θ_1 ;z_2=r_2e^i^θ ...

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Byju's Answer

Standard XII

Mathematics

Argument of a Complex Number

If |z1+z2|=|z...

Question

If | $z_{1}$ + $z_{2}$ | =| $z_{1}$ - $z_{2}$ | , then arg $z_{1}$ - arg $z_{2}$ =

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Solution

The correct option is C

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

$\Rightarrow$ $z_{1}$ . ${¯ ¯ ¯ z}_{2}$ + ${¯ ¯ ¯ z}_{1}$ . $z_{2}$ =0

i.e, $z_{1}$ . ${¯ ¯ ¯ z}_{2}$ + $¯ ¯¯¯¯¯¯¯¯¯¯ ¯ z_{1} . {¯ ¯ ¯ z}_{2}$ =0

$\Rightarrow$ Re( $z_{1}$ . ${¯ ¯ ¯ z}_{2}$ ) =0

Put $z_{1}$ = $r_{1}$ $e^{i}$ $^{θ_{1}}$ ; $z_{2}$ = $r_{2}$ $e^{i}$ $^{θ_{2}}$

$\Rightarrow$ Re $z_{1}$ . ${¯ ¯ ¯ z}_{2}$ = $r_{1}$ $r_{2}$ $e^{i (θ_{1} - θ_{2})}$ =0

$\Rightarrow$ cos( $θ_{1} - θ_{2})$ =0

$\Rightarrow$ $θ_{1} - θ_{2}$ = $\pm \frac{π}{2}$

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If |z1 + z2| =|z1 - z2| , then arg z1 - arg z2 =

The correct option is C |z1+z2|2 = |z1−z2|2 ⇒ z1.¯¯¯z2 + ¯¯¯z1.z2 =0 i.e, z1.¯¯¯z2 + ¯¯¯¯¯¯¯¯¯¯¯¯z1.¯¯¯z2 =0 ⇒ Re(z1.¯¯¯z2) =0 Put z1=r1eiθ1 ;z2=r2eiθ2 ⇒ Re z1.¯¯¯z2 = r1r2ei(θ1−θ2) =0 ⇒ cos(θ1−θ2)=0 ⇒ θ1−θ2 =±π2

If | $z_{1}$ + $z_{2}$ | =| $z_{1}$ - $z_{2}$ | , then arg $z_{1}$ - arg $z_{2}$ =