JEE Advanced 2025 · Paper 1 · Mathematics Q7

MathematicsMultiple correct+4 / -2Question 7 of 48

Let $\mathbb{R}$ denote the set of all real numbers. Let $z_1 = 1 + 2i$ and $z_2 = 3i$ be two complex numbers, where $i = \sqrt{-1}$ . Let

$S = \{(x, y) \in \mathbb{R}\times\mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2|\}.$

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

S

is a circle with centre

\left(-\dfrac13, \dfrac{10}{3}\right)

S

is a circle with centre

\left(\dfrac13, \dfrac83\right)

S

is a circle with radius

\dfrac{\sqrt2}{3}

S

is a circle with radius

\dfrac{2\sqrt2}{3}

Answer and worked solution

Official answer: A, D

Write $z=x+iy$ . Square $|z-z_1|=2|z-z_2|$ :
$(x-1)^2+(y-2)^2=4\left(x^2+(y-3)^2\right)\implies x^2+y^2+\tfrac23x-\tfrac{20}3y+\tfrac{31}3=0.$

Centre. $\left(-\frac13,\frac{10}3\right)$ .

Radius.
$r^2=\frac19+\frac{100}9-\frac{31}3=\frac89\implies r=\frac{2\sqrt2}{3}.$

JEE Advanced 2025 · Paper 1, Mathematics question 7