Let z be a complex number such that z - i z + 2 i = 1 and z = 5 2 . Then, the value of z + 3 i is

Given, z - i = z + 2 i ⇒ x 2 + y - 1 2 = x 2 + y + 2 2 ⇒ - 2 y + 1 = 4 y + 4 ⇒ 6 y = - 3 ⇒ y = - 1 2 From, z = 5 2 , we get, x 2 + y 2 = 25 4 ⇒ x 2 = 24 4 = 6 ⇒ z = ± 6 - i 2 Hence, z + 3 i = 6 + 25 4 = 49 4 z + 3 i = 7 2

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Mathematics - Complex Number Question with Solution | TestHub

MathematicsComplex NumberGeometric ApplicationMedium2 minPYQ_2020

MathematicsMediumsingle choice

Let $z$ be a complex number such that $|\frac{z - i}{z + 2 i}| = 1$ and $|z| = \frac{5}{2}$ . Then, the value of $|z + 3 i|$ is

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Solution:

Given, $|z - i| = |z + 2 i|$

$\Rightarrow x^{2} + {(y - 1)}^{2} = x^{2} + {(y + 2)}^{2}$
$\Rightarrow - 2 y + 1 = 4 y + 4$
$\Rightarrow 6 y = - 3 \Rightarrow y = - \frac{1}{2}$

From, $|z| = \frac{5}{2}$ , we get,
$x^{2} + y^{2} = \frac{25}{4} \Rightarrow x^{2} = \frac{24}{4} = 6 \Rightarrow z = \pm \sqrt{6} - \frac{i}{2}$
Hence, $|z + 3 i| = \sqrt{6 + \frac{25}{4}} = \sqrt{\frac{49}{4}}$
$|z + 3 i| = \frac{7}{2}$

Stream:JEESubject:MathematicsTopic:Complex NumberSubtopic:Geometric Application

⏱ 2mℹ️ Source: PYQ_2020

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