First slide

Algebra of complex numbers

Question

If $x + i y = \frac{1}{1 - \cos θ + 2 i \sin θ}, θ \neq 2 n π, n \in I$ then maximum value of x is

Moderate

A

1
B

2
C

$\frac{1}{2}$
D

$\frac{1}{3}$

Solution

$2 x = (x + i y) + (x - i y) = \frac{2 (1 - \cos θ)}{(1 - \cos θ)^{2} + 4 \sin^{2} θ}$
$\Rightarrow x = \frac{1}{5 + 3 \cos θ}$
Thus, maximum value of $x$ is $\frac{1}{2}$ . It is attained at $θ = π$ .

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