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

Correct answer: B. (7)/(2) Given, |z - i| = |z + 2i| ⇒ x ^ 2 + (y - 1) ^ 2 = x ^ 2 + (y + 2) ^ 2 ⇒ - 2y + 1 = 4y + 4 ⇒ 6y = - 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 + 3i| = 6 + (25)/(4) = (49)/(4) |z + 3i|=(7)/(2)

JEE Main 20209 JanuaryMorning Shift

Exam: JEE Main
Year: 2020
Date: 9 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Complex Number

JEE Main 2020 · 9 January · Morning Shift · Mathematics · Complex Number

Let z be a complex number such that |(z - i)/(z + 2i)| = 1 and |z|...

Let

z be a complex number such that \left|\dfrac{z - i}{z + 2i}\right| = 1

and \left|z\right| = \dfrac{5}{2}

. Then, the value of \left|z + 3i\right|

Options

A $\sqrt{10}$
B $\dfrac{7}{2}$
C $\dfrac{15}{4}$
D $2\sqrt{3}$

Correct answer

B. $\dfrac{7}{2}$

Step-by-step solution

Given, $\left|z - i\right| = \left|z + 2i\right|$ ⇒ {x}^{2} + {\left(y - 1\right) $}^{2} = {x}^{2} + {\left(y + 2\right)$ }^{2} $ $ ⇒ - 2y + 1 = 4y + 4 $ $ ⇒ 6y = - 3 ⇒ y = - $\dfrac{1}{2}From, \left|z\right| = \dfrac{5}{2}, we get, $ {x}^{2} + {y}^{2} = $\dfrac{25}{4}$ ⇒ {x}^{2} = $\dfrac{24}{4} =$ 6 ⇒ z = \pm \sqrt{6} - $\dfrac{i}{2} Hence, \left|z + 3i\right| = \sqrt{6 + \dfrac{25}{4}} = \sqrt{\dfrac{49}{4}}$
\left|z + 3i\right|=\dfrac{7}{2}$

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