If a > 0 and z = ( 1 + i ) 2 a - i , has magnitude 2 5 , then z - is equal to:

Z = 1 + i 2 a - i = 2 i a - i z = 2 5 ⇒ 2 i a - i = 2 a 2 + 1 = 2 5 ⇒ a = 3 ∵ a > 0 So, z - = 2 i 3 - i - = - 2 i 3 + i = - 2 i 3 + i × 3 - i 3 - i = - 6 i - 2 3 2 + 1 = - 1 5 - 3 5 i

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Mathematics - Complex Number Question with Solution | TestHub

MathematicsComplex NumberGeneral(Modulus,Argument,Conjugate)Medium2 minPYQ_2019

MathematicsMediumsingle choice

If $a > 0$ and $z = \frac{{(1 + i)}^{2}}{a - i}$ , has magnitude $\sqrt{\frac{2}{5}}$ , then $\bar{z}$ is equal to:

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Answer:

Solution:

$Z = \frac{{(1 + i)}^{2}}{a - i} = \frac{2 i}{a - i}$
$|z| = \sqrt{\frac{2}{5}} \Rightarrow |\frac{2 i}{a - i}| = \frac{2}{\sqrt{a^{2} + 1}} = \sqrt{\frac{2}{5}}$
$\Rightarrow a = 3 (∵ a > 0)$
So, $\bar{z} = \bar{(\frac{2 i}{3 - i})} = \frac{- 2 i}{3 + i} = \frac{- 2 i}{3 + i} \times \frac{3 - i}{3 - i}$
$= \frac{- 6 i - 2}{3^{2} + 1} = - \frac{1}{5} - \frac{3}{5} i$

Stream:JEESubject:MathematicsTopic:Complex NumberSubtopic:General(Modulus,Argument,Conjugate)

⏱ 2mℹ️ Source: PYQ_2019

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