JEE Advanced 2024 — Paper 1 — MATH-II solution

JEE Advanced2024Paper 1MATH-II

Let $R^{3}$ denote the three-dimensional space. Take two points $P = (1, 2, 3)$ and $Q = (4, 2, 7)$ . Let $dist (X, Y)$ denote the distance between two points $X$ and $Y$ in $R^{3}$ . Let

$S = {X \in R^{3} : (dist (X, P))^{2} - (dist (X, Q))^{2} = 50}$ and

$T = {Y \in R^{3} : (dist (Y, Q))^{2} - (dist (Y, P))^{2} = 50}$ .

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

A
There is a triangle whose area is 1 and all of whose vertices are from $S$ .
B
There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $L M$ is also in $T$ .
C
There are infinitely many rectangles of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$ .
D
There is a square of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$ .

Solution

Expanding $(dist (X, P))^{2} - (dist (X, Q))^{2}$ with $X = (x, y, z)$ gives a linear expression. The set $S$ becomes the plane $6 x + 8 z - 105 = 0$ , and similarly $T$ is the plane $6 x + 8 z - 5 = 0$ .

(A) A plane contains triangles of any positive area, including area 1. True.

(B) Any two points of a plane have the line segment joining them lying in the plane. True.

(C) The two planes are parallel. The perpendicular distance between them equals $\frac{∣105 - 5∣}{6 ^{2} + 8 ^{2}} = \frac{100}{10} = 10$ . A rectangle with two vertices on each plane has one pair of opposite sides of length $10$ (the perpendicular distance) and the other pair of length $\frac{48 - 2 \cdot 10}{2} = 14$ . Such rectangles can be rotated within the planes, giving infinitely many. True.

(D) A square would require all sides equal to $12$ , but the inter-plane distance is fixed at $10 \neq = 12$ . False.

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