If the sum of squares of all real values of α, for which the lines 2x-y+3=0,6x+3y+1=0 and α x+2y-2=0 do not form a triangle is p, then the greatest integer less than or equal to p is _________.

Solution

Maths sir · 4 steps

No triangle means two lines are parallel or all three meet at one point, so catch every such α.

1. When no triangle. Three lines do not form a triangle if any two are parallel, or if all three pass through one point. The first two lines have slopes 2 and -2, so they always cross. The third line has slope -α/2.
2. Parallel cases. Parallel to 2x-y+3=0: -α/2 = 2, so α = -4. Parallel to 6x+3y+1=0: -α/2 = -2, so α = 4.
3. Concurrent case. Put y = 2x+3 into 6x+3y+1=0: 12x+10=0, so x = -5/6 and y = 4/3. The third line passes through this point when -5α/6 + 8/3 - 2 = 0, which gives α = 4/5.

   p = (-4)² + 4² + (4/5)² = 16 + 16 + 16/25 = 32.64

4. So. The greatest integer less than or equal to 32.64 is 32.

The common slip: 16. Counting only one of the two parallel cases gives p = 16.64, so the answer comes out as 16. The third line can be parallel to either of the first two.

The idea to keep. For "lines do not form a triangle", always collect both kinds of values: one for each line it can be parallel to, and one for the common point of the other two.

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