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=√(-1) . Let S= \ (x, y) in mathbb R × mathbb R : x+i y-z 1 =2 x+i y-z 2 \ . Then which of the following statements is (are) TRUE ?

S is a circle with centre (-1/3, 10/3 )

S is a circle with radius frac 2 √(2) 3

S is a circle with centre (1/3, 8/3 )

S is a circle with radius frac √(2) 3

Mathematics · Complex Numbers

JEE Advanced 2025 — Paper 1 — Question 21

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 ?

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

$\quad|x+i y-1-2 i|=2|x+i y-3 i|$

$\Rightarrow \quad(\mathrm{x}-1)^{2}+(\mathrm{y}-2)^{2}=4\left(\mathrm{x}^{2}+(\mathrm{y}-3)^{2}\right)$

$\Rightarrow \quad 3 x^{2}+3 y^{2}+2 x-20 y+31=0$

$\Rightarrow \quad x^{2}+y^{2}+\frac{2 x}{3}-\frac{20 y}{3}+\frac{31}{3}=0$

$\therefore \mathrm{S}$ is a circle with centre $\left(-\frac{1}{3}, \frac{10}{3}\right)$ and radius

$=\sqrt{\frac{1}{9}+\frac{100}{9}-\frac{31}{3}}=\sqrt{\frac{8}{9}}=\frac{2 \sqrt{2}}{3}$

Answer key and solution verified before publishing.

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