The lines L₁, L₂,…,L₂₀ are distinct. For n=1,2,3,…,10 all the lines L₂ₙ₋₁ are parallel to each other and all the lines L₂ₙ pass through a given point P. The maximum number of points of intersection of pairs of lines from the set {L₁, L₂,…,L₂₀} is equal to ___________.

Solution

Maths sir · 3 steps

Start from all 190 pairs, then remove what parallel lines lose and what concurrent lines share.

1. Two families. The odd-numbered lines L₁, L₃, …, L₁₉ are 10 parallel lines. The even-numbered lines L₂, L₄, …, L₂₀ are 10 lines through the same point P.

   Pairs taken from	Number of pairs	Points of intersection
   parallel with parallel	C(10, 2) = 45	0, parallel lines never meet
   concurrent with concurrent	C(10, 2) = 45	1, all of them meet only at P
   parallel with concurrent	10 × 10 = 100	100, all different points

2. Making it maximum. The 100 cross points are all different when no concurrent line is parallel to the family, no parallel line passes through P, and no two cross points coincide. Two concurrent lines already meet at P, so they cannot meet the same parallel line at the same point either.

   0 + 1 + 100 = 101

3. So. As a check, C(20, 2) - 45 - 45 + 1 = 190 - 90 + 1 = 101. The maximum number of points of intersection is 101.

The common slip: 145. Counting the 45 pairs of concurrent lines as 45 separate points gives 145, but all of those pairs meet at the one point P.

The idea to keep. For maximum intersections, count all pairs with C(n, 2). Then remove every pair of parallel lines, and replace C(m, 2) by 1 for each set of m concurrent lines.

Answer from the NTA final answer key. Solution by PrepWiser.