JEE Advanced 2025 — Paper 1 — MATH-II solution

JEE Advanced2025Paper 1MATH-II

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

$S = {(x, y) \in R \times R : ∣ x + i y - z_{1} ∣ = 2∣ x + i y - z_{2} ∣} .$

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

Solution

Square the defining condition $∣ x + i y - (1 + 2 i) ∣ = 2∣ x + i y - 3 i ∣$ :

$(x - 1)^{2} + (y - 2)^{2} = 4 (x^{2} + (y - 3)^{2}) .$

Expanding both sides:

$x^{2} - 2 x + 1 + y^{2} - 4 y + 4 = 4 x^{2} + 4 y^{2} - 24 y + 36.$

Bringing all terms to one side and dividing by $3$ :

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

This is a circle with centre $(- \frac{1}{3}, \frac{10}{3})$ and radius

$r = \frac{1}{9} + \frac{100}{9} - \frac{31}{3} = \frac{101 - 93}{9} = \frac{8}{9} = \frac{2 2}{3} .$

$∴$ The correct options are (A) and (D).

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