The lines are distinct. For all the lines are parallel to each other and all the lines pass through a given point P . The maximum number of points of intersection of pairs of lines from the set is equal to :
Answer: 101
Solution
There are lines: of them () are mutually parallel, and the other () all pass through one common point .
Total pairs of lines: .
Pairs among the parallel lines never intersect, so remove points.
Pairs among the concurrent lines all meet at the single point — that's pairs collapsing into just distinct intersection point instead of distinct points, a reduction of .
The remaining pairs — one line from each family — number , and in the maximal (general position) case each gives a distinct point of intersection.
Total distinct points .
(Equivalently: points from mixed pairs point from the concurrent family .)
Total pairs of lines: .
Pairs among the parallel lines never intersect, so remove points.
Pairs among the concurrent lines all meet at the single point — that's pairs collapsing into just distinct intersection point instead of distinct points, a reduction of .
The remaining pairs — one line from each family — number , and in the maximal (general position) case each gives a distinct point of intersection.
Total distinct points .
(Equivalently: points from mixed pairs point from the concurrent family .)
JEE Main · 2024 · 01 Feb 2024 · Shift 2 · Integer answer