UPSC NDA II 2024 Mathematics Paper

Concept:The equation of a circle can be rewritten in factorized form to identify points that lie on a diameter.Explanation:Start with the given equation: $(x^2 - 4x + 3) + (y^2 - 6y + 8) = 0$Factorize each quadratic expression:$x^2 - 4x + 3 = (x-3)(x-1)$ $y^2 - 6y + 8 = (y-4)(y-2)$ Thus the equation becomes $(x-3)(x-1) + (y-4)(y-2) = 0$.This form suggests that the endpoints of a diameter could be the points making each factor zero: $(3,4), (3,2), (1,4), (1,2)$.Complete the square to find the center: $(x^2 - 4x + 4) + (y^2 - 6y + 9) = -3 - 8 + 4 + 9$ $(x-2)^2 + (y-3)^2 = 2$ So the center is $(2,3)$ and radius $\sqrt{2}$. For a diameter, the endpoints must be symmetric about the center: Check pair I: $(1,2)$ and $(3,4)$ — midpoint is $(2,3)$, distance $= 2\sqrt{2} = 2 \times$ radius. Correct. Check pair II: $(1,4)$ and $(3,2)$ — midpoint is $(2,3)$, distance $= 2\sqrt{2}$. Correct. Check pair III: $(2,4)$ and $(4,2)$ — midpoint is $(3,3)$, not the center. Incorrect.Answer:Therefore, statements I and II are correct, which corresponds to option A.