Question:
If the conjugate of a complex number $z$ is $\frac{1}{i - 1}$ , then $z$ is
If the conjugate of a complex number $z$ is $\frac{1}{i - 1}$ , then $z$ is
Updated On: Jun 7, 2024
- $\frac{1}{i - 1} $
- $\frac{1}{i + 1} $
- $\frac{-1}{i - 1} $
- $\frac{ - 1}{i + 1} $
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The Correct Option is D
Solution and Explanation
$z=\frac{1}{i-1} \times \frac{i+1}{i+1}$
$\Rightarrow \, z=\frac{i+1}{i^{2}-1^{2}}$
$z=-\frac{1}{2} \times(i+1)$
$\Rightarrow \, \bar{z}=-\frac{1}{2}(1-i) \times \frac{(1+i)}{(1+i)}$
$=-\frac{1}{2} \frac{(1+1)}{(1+i)}=-\frac{1}{(1+i)}$
$\Rightarrow \, z=\frac{i+1}{i^{2}-1^{2}}$
$z=-\frac{1}{2} \times(i+1)$
$\Rightarrow \, \bar{z}=-\frac{1}{2}(1-i) \times \frac{(1+i)}{(1+i)}$
$=-\frac{1}{2} \frac{(1+1)}{(1+i)}=-\frac{1}{(1+i)}$
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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.