Let P_1, P_2, ..., P_k be points in the xy-plane such that no three of them are collinear. For every pair of points P_i and P_j, let L_{ij} be the line passing through them. Let L_{ab} be the line with the steepest gradient amongst all lines L_{ij}.
Which one of the following properties should necessarily be satisfied?
A. P_a and P_b are adjacent to each other with respect to their x-coordinates.
B. Either P_a or P_b has the largest or smallest y-coordinate amongst all the points.
C. The difference between the y-coordinates of P_a and P_b is minimum.
D. None of the above.
GATE 2007 · Engineering Mathematics · Cartesian Coordinates · medium
Answer: A. P_a and P_b are adjacent to each other with respect to their x-coordinates.
Set up slope formula and assume non-adjacency: Suppose P_a and P_b are not x-adjacent. Then there exists P_c with x_a < x_c < x_b. Let s = (y_b - y_a)/(x_b - x_a) be the slope of L_{ab}.
Show x-adjacent pair has slope >= s: The slope s is a weighted average: s = [(x_c - x_a)*s_{ac} + (x_b - x_c)*s_{cb}] / (x_b - x_a). So max(s_{ac}, s_{cb}) >= s. This means L_{ab} is not the steepest line, a contradiction.
Conclude option A is necessary: By contradiction, P_a and P_b must be adjacent with respect to their x-coordinates. Options B, C, D can be ruled out by simple counterexamples.