Question:
If $ \sqrt{x+iy}=\pm (a+ib), $ then $ \sqrt{-x-iy} $ is equal to
If $ \sqrt{x+iy}=\pm (a+ib), $ then $ \sqrt{-x-iy} $ is equal to
Updated On: Jun 8, 2024
- $ \pm (b+ia) $
- $ \pm (a-ib) $
- $ (ai+b) $
- $ \pm (b-ia) $
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The Correct Option is D
Solution and Explanation
Given, $ \sqrt{x+iy}=\pm (a+bi) $
$ \Rightarrow $ $ x+iy={{a}^{2}}-{{b}^{2}}+2iab $
$ \Rightarrow $ $ x={{a}^{2}}-{{b}^{2}},y=2ab $
$ \therefore \,\,\,\,\sqrt{-x-iy}=\,\sqrt{-({{a}^{2}}-{{b}^{2}})-2\,iab} $
$=\sqrt{{{b}^{2}}-{{a}^{2}}-2iab}\,=\sqrt{{{(b-ia)}^{2}}} $
$=\pm \,(b-ia) $
$ \Rightarrow $ $ x+iy={{a}^{2}}-{{b}^{2}}+2iab $
$ \Rightarrow $ $ x={{a}^{2}}-{{b}^{2}},y=2ab $
$ \therefore \,\,\,\,\sqrt{-x-iy}=\,\sqrt{-({{a}^{2}}-{{b}^{2}})-2\,iab} $
$=\sqrt{{{b}^{2}}-{{a}^{2}}-2iab}\,=\sqrt{{{(b-ia)}^{2}}} $
$=\pm \,(b-ia) $
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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.