Let z 1 and z 2 be two complex numbers such that $z_1=iz_2 \,and \,arg(\frac{z_1}{z_2})=π.$

The correct option is: (C). ∵ $\frac{z_1}{z_2}=i⇒z_1=-iz_2$ $⇒arg(z_1)=-\frac{\pi}{2}+arg(z_2).....(i)$ Also arg(z1)-arg(z2)= π ⇒ arg( z 1 ) + arg( z 2 ) = π …(ii) From (i) and (ii), we get $arg(z_1)=\frac{\pi}{4} and\,arg(z_2)=\frac{3\pi}{4}$

$arg \,z_1=-(\frac{3\pi}{4})$

Let z 1 and z 2 be two complex numbers such that $z_1=iz_2 \,and \,arg(\frac{z_1}{z_2})=π.$

$arg \,z_1=-(\frac{3\pi}{4})$

Question:

Let z₁ and z₂ be two complex numbers such that
$z_1=iz_2 \,and \,arg(\frac{z_1}{z_2})=π.$

Updated On: Mar 4, 2024

$arg \,z_2=(\frac{\pi}{4})$
$arg \,z_2=-(\frac{3\pi}{4})$
$arg \,z_1=(\frac{\pi}{4})$
$arg \,z_1=-(\frac{3\pi}{4})$

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The Correct Option is A

Solution and Explanation

The correct option is: (C).

∵ $\frac{z_1}{z_2}=i⇒z_1=-iz_2$

$⇒arg(z_1)=-\frac{\pi}{2}+arg(z_2).....(i)$

Also

arg(z1)-arg(z2)= π

⇒ arg(z₁) + arg(z₂) = π …(ii)

From (i) and (ii), we get

$arg(z_1)=\frac{\pi}{4} and\,arg(z_2)=\frac{3\pi}{4}$

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Concepts Used:

Complex Number

A Complex Number is written in the form

a + ib

where,

“a” is a real number
“b” is an imaginary number

The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.

Let z1 and z2 be two complex numbers such that z1=iz2 and arg(z1z2)=π.z_1=iz_2 \,and \,arg(\frac{z_1}{z_2})=π.z1​=iz2​andarg(z2​z1​​)=π.