Let - and - be two distinct complex numbers such that - - - If real part of - is positive and imaginary part of - is negative- then the complex number - - - - - - may be

The correct option is C purely imaginaryLet Z= #160; α +β #160;α β #160; #160; cis(θ )=cosθ #160; +isinθ #160; #160; Let, | ...

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Byju's Answer

Standard X

Mathematics

Trigonometric Ratios of Complementary Angles

Let α and ...

Question

Let $α$ and $β$ be two distinct complex numbers such that $| α | = | β |$ . If real part of $α$ is positive and imaginary part of $β$ is negative, then the complex number $\frac{(α + β)}{(α - β)}$ may be

zero

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real and negative

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real and positive

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purely imaginary

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Solution

The correct option is C purely imaginary
Let $Z =$ $\frac{α + β}{α - β}$
$c i s (θ) = cos θ + i sin θ$
Let, $| α | = | β | = r & α = r c i s (A), β = r c i s (B)$
$Z = \frac{c i s (A) + c i s (B)}{c i s (A) - c i s (B)}$
$=$ $\frac{cos A + cos B + i (sin A + sin B)}{cos A + cos B - i (sin A + sin B)}$
$=$ $\frac{cos (\frac{A - B}{2}) [cos (\frac{A + B}{2}) + i sin (\frac{A + B}{2})]}{- sin (\frac{A - B}{2}) [sin (\frac{A + B}{2}) - i cos (\frac{A + B}{2})]}$
$Z = \frac{- i cos (\frac{A - B}{2})}{sin (\frac{A - B}{2})}$
Therefore, $Z$ is purely imaginary.

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Let α and β be two distinct complex numbers such that |α|=|β|. If real part of α is positive and imaginary part of β is negative, then the complex number (α+β)(α−β) may be

Let $α$ and $β$ be two distinct complex numbers such that $| α | = | β |$ . If real part of $α$ is positive and imaginary part of $β$ is negative, then the complex number $\frac{(α + β)}{(α - β)}$ may be