Let P1, P2,..... , Pn be n points in the xy-plane such that no three of them…
2007
Let P1, P2,..... , Pn be n points in the xy-plane such that no three of them are collinear. For every pair of points Pi and Pj, let Lij be the line passing through them. Let Lab be the line with the steepest gradient amongst all n(n -1)/2 lines.
Which one of the following properties should necessarily be satisfied ?
- A.
Pa and Pb are adjacent to each other with respect to their x-coordinate
- B.
Either Pa or Pb has the largest or the smallest y-coordinate among all the points
- C.
The difference between x-coordinatef Pa and Pb is minimum
- D.
None of the above
Show answer & explanation
Correct answer: D
To determine the line with the steepest gradient, consider that the gradient of a line between two points (x₁, y₁) and (x₂, y₂) is given by (y₂ - y₁)/(x₂ - x₁). The steepest gradient occurs when the absolute value of this ratio is maximized.
Step 1: A large gradient requires a large change in y relative to a small change in x. This means that even if two points are close in x, if their y-values differ significantly, the gradient can be steep.
Step 2: Consider option A: Adjacency in x-coordinate does not guarantee a large y-difference. For example, two points with very close x-values but nearly identical y-values will have a small gradient. Thus, this property is not necessarily true.
Step 3: Consider option B: If one point has the maximum or minimum y-coordinate, then the line connecting it to another point may have a large y-difference. However, this is not guaranteed unless the x-difference is also small. So this property is not necessarily true either.
Step 4: Consider option C: A minimum x-difference does not ensure a steep gradient if the y-difference is small. For example, two points with very close x-values but nearly equal y-values will have a small gradient. Thus, this property is not necessarily true.
Step 5: Since none of the options A, B, or C are necessarily true, the correct choice is D: None of the above.