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AP EAMCET Engineering 17 Sep 2020 Shift 2 Paper

Section: Mathematics

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Question : 21 of 160

Marks: +1, -0

Let

z_1, z_2

be two complex number such that

\overline{z}_1 - \overline{iz}_2 = 0

\arg(z_1 z_2) = \frac{3\pi}{4},

\arg(z_1) =

$\frac{\pi}{4}$
$\frac{-\pi}{8}$
$\frac{\pi}{8}$
$\frac{\pi}{3}$

Solution:

Given

\overline{z}_1 - i \overline{z}_2 = 0

\Rightarrow \overline{z}_1 = i \overline{z}_2

\Rightarrow \overline{z}_1 = \overline{i z_2}

\Rightarrow z_1 = -i z_2

Clearly argument of

z_1 =

argument of

z_2 - \frac{\pi}{2}

or

argument

z_1 =

argument

z_2 - \frac{\pi}{2}

\Rightarrow

argument

z_2 =

argument

z_1 + \frac{\pi}{2}

Let argument

z_1 = \alpha

Then, given argument

(z_1 z_2) = \frac{3\pi}{4}

\Rightarrow

argument

z_1 +

argument

z_2 = \frac{3\pi}{4}

\alpha + \alpha + \frac{\pi}{2} = \frac{3\pi}{4}

2\alpha = \frac{3\pi}{4} - \frac{\pi}{2} \Rightarrow \alpha = \frac{\pi}{8}

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