Let z 1 and z 2 be two complex numbers such that z 1- i z 2-0- z 1 z 2- - Then which of the following is-are tureA- z 1-0- 5 -6B- z 1-0- -2C- z 2-3- 2 -3D- z 2-0- -2

Given: (z_1 z_2)=π, z_1+i z_2=0 Now, z_1+i z_2=0 ⇒z_1= i z_2 ⇒z_1= i z_2 ⇒ z_1=i z_2 ⇒(z_1)=(i z_2) ⇒ z_1= i+ z_2 ⇒ z_1 z_2=π2 ...(1) (z_1 z_2)=π ⇒ n ...

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Byju's Answer

Standard XII

Mathematics

Properties of Argument

Let z 1 and z...

Question

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $_{1} + i_{2} = 0, arg (z_{1} z_{2}) = π . Then$ which of the following is/are ture

arg z_{1} \in (0, \frac{5 π}{6})

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arg z_{1} \in (0, \frac{π}{2})

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arg z_{2} \in (\frac{π}{3}, \frac{2 π}{3})

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arg z_{2} \in (0, \frac{π}{2})

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Solution

The correct options are
A $arg z_{1} \in (0, \frac{5 π}{6})$
D $arg z_{2} \in (0, \frac{π}{2})$
$_{1} + i_{2} = 0$
$\Rightarrow_{1} = - i_{2}$
$\Rightarrow ¯ ¯¯¯ ¯_{1} = ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ - i_{2}$
$\Rightarrow z_{1} = i z_{2}$
$\Rightarrow arg (z_{1}) = arg (i z_{2})$
$\Rightarrow arg z_{1} = arg i + arg z_{2}$
$\Rightarrow arg z_{1} - arg z_{2} = \frac{π}{2} \dots (1)$
$∵ arg z_{1} + arg z_{2} = π \dots (2)$
By equations $(1)$ and $(2)$
$2 arg z_{1} = \frac{3 π}{2} \Rightarrow arg z_{1} = \frac{3 π}{4}$
and $arg z_{2} = \frac{π}{4}$

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Let z1 and z2 be two complex numbers such that ¯¯¯¯¯z1+i ¯¯¯¯¯z2=0,arg(z1z2)=π. Then which of the following is/are ture

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $_{1} + i_{2} = 0, arg (z_{1} z_{2}) = π . Then$ which of the following is/are ture