Question:
If $z = \frac{4}{1-i}$, then $\bar{z}$ is (where $\bar{z}$ is complex conjugate of $z$ )
If $z = \frac{4}{1-i}$, then $\bar{z}$ is (where $\bar{z}$ is complex conjugate of $z$ )
Updated On: Jun 18, 2022
- $2 (1 + i)$
- $(1 + i)$
- $\frac{2}{1-i}$
- $\frac{4}{1+i}$
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The Correct Option is D
Solution and Explanation
$z=\frac{4}{1-i} $
$\Rightarrow z=\frac{4}{1-i} \times \frac{1+i}{1+i}$
$=\frac{4(1+i)}{1^{2}-i^{2}}=2(1+i)$
$\therefore \bar{z}=\frac{2(1-i) \times(1+i)}{1+i}$
$=\frac{4}{1+i}$
$\Rightarrow z=\frac{4}{1-i} \times \frac{1+i}{1+i}$
$=\frac{4(1+i)}{1^{2}-i^{2}}=2(1+i)$
$\therefore \bar{z}=\frac{2(1-i) \times(1+i)}{1+i}$
$=\frac{4}{1+i}$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
