Let ℝ denote the set of all real numbers. Let z_1=1+2 i and z_2=3 i be two complex numbers, where i= -1 . Let S=\ (x, y) ∈ ℝ × ℝ:|x+i y-z_1|=2|x+i y-z_2|\ . Then which of the following statements is (are) TRUE?

Correct answer: A. S is a circle with centre (-(1)/(3), (10)/(3)) aligned & aligned & |x+i y-1-2 i|=2|x+i y-3 i| & (x-1)^2+(y-2)^2=4(x^2+(y-3)^2) & 3 x^2+3 y^2+2 x-20 y+31=0 & x^2+y^2+(2 x)/(3)-(20 y)/(3)+(31)/(3)=0 aligned & S is a circle with centre (-(1)/(3), (10)/(3)) and radius = (1)/(9)+(100)/(9)-(31)/(3) = (8)/(9) = 2 2 3 aligned

JEE Advanced 2025Paper 1

Exam: JEE Advanced
Year: 2025
Paper: Paper 1
Subject: Mathematics
Chapter: Complex Number

JEE Advanced 2025 · Paper 1 · Mathematics · Complex Number

Let ℝ denote the set of all real numbers. Let z_1=1+2 i and z_2=3 i...

Let

\mathbb{R}

denote the set of all real numbers. Let

z_1=1+2 i

and

z_2=3 i

be two complex numbers, where

i=\sqrt{-1}

. Let

S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}:\left|x+i y-z_1\right|=2\left|x+i y-z_2\right|\right\}

. Then which of the following statements is (are) TRUE?

Options

A $S$ is a circle with centre $\left(-\frac{1}{3}, \frac{10}{3}\right)$
B $S$ is a circle with centre $\left(\frac{1}{3}, \frac{8}{3}\right)$
C $S \text{ is a circle with radius } \frac{\sqrt{2}}{3}$
D $S \text{ is a circle with radius } \frac{2 \sqrt{2}}{3}$

Correct answer

A. $S$ is a circle with centre $\left(-\frac{1}{3}, \frac{10}{3}\right)$

Step-by-step solution

\begin{aligned} &\begin{aligned} & |x+i y-1-2 i|=2|x+i y-3 i| \\ & \Rightarrow \quad(x-1)^2+(y-2)^2=4\left(x^2+(y-3)^2\right) \\ & \Rightarrow \quad 3 x^2+3 y^2+2 x-20 y+31=0 \\ & \Rightarrow \quad x^2+y^2+\frac{2 x}{3}-\frac{20 y}{3}+\frac{31}{3}=0 \end{aligned}

\\ &\therefore \mathrm{S} \text{ is a circle\text{ with } centre }\left(-\frac{1}{3}, \frac{10}{3}\right) \text{ and radius }=\sqrt{\frac{1}{9}+\frac{100}{9}-\frac{31}{3}}=\sqrt{\frac{8}{9}}=\frac{2 \sqrt{2}}{3} \end{aligned}$

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