Let z ∈ C, the set of complex numbers. Then the equation, 2|z + 3i| - |z - i| = 0 represents:

Correct answer: A. A circle with radius (8)/(3) Let z = x + iy so that 2|x + i(y + 3)| = |x + i(y - 1)| ⇒ 2 x ^ 2 + (y + 3) ^ 2 = x ^ 2 + (y - 1) ^ 2 ⇒ 4 x ^ 2 + 4 (y + 3) ^ 2 = x ^ 2 + (y - 1) ^ 2 ⇒ 3 x ^ 2 + 3 y ^ 2 +26 y + 35 = 0 ⇒ x ^ 2 + y ^ 2 + (26)/(3)y + (35)/(3) = 0 ⇒ r = 0 + (169)/(9) - (35)/(3) ⇒ r = (64)/(9) = (8)/(3)

JEE Main 20178 AprilOnline

Exam: JEE Main
Year: 2017
Date: 8 April
Mode: Online
Subject: Mathematics
Chapter: Complex Number

JEE Main 2017 · 8 April · Online · Mathematics · Complex Number

Let z ∈ C, the set of complex numbers. Then the equation, 2|z + 3i|...

Let

z \in C,

the set of complex numbers. Then the equation,

2\left|z + 3i\right| - \left|z - i\right| = 0

represents:

Options

AA circle with radius $\dfrac{8}{3}$
BAn ellipse with length of minor axis $\dfrac{16}{9}$
CAn ellipse with length of major axis $\dfrac{16}{3}$
DA circle with diameter $\dfrac{10}{3}$

Correct answer

A. A circle with radius $\dfrac{8}{3}$

Step-by-step solution

Let

z = x + iy

so that

2\left|x + i\left(y + 3\right)\right| = \left|x + i\left(y - 1\right)\right|

⇒ 2\sqrt{{x}^{2} + {\left(y + 3\right)}^{2}}=\sqrt{{x}^{2} + {\left(y - 1\right)}^{2}}

⇒ 4{x}^{2} + 4{\left(y + 3\right)}^{2} = {x}^{2} + {\left(y - 1\right)}^{2}

⇒ 3{x}^{2} + 3{y}^{2} +26 y + 35 = 0

⇒ {x}^{2} + {y}^{2} + \dfrac{26}{3}y + \dfrac{35}{3} = 0

⇒ r = \sqrt{0 + \dfrac{169}{9} - \dfrac{35}{3}}

⇒ r = \sqrt{\dfrac{64}{9}} = \dfrac{8}{3}

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