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

Step-by-step solution

Given lines are 2 x - y + 3 = 0 , 6 x + 3 y + 1 = 0 and αx + 2 y - 2 = 0 . These lines will not form a triangle if αx + 2 y - 2 = 0 is concurrent with 2 x - y + 3 = 0 and 6 x + 3 y + 1 = 0 or parallel to either of them. Case- 1 : Concurrent lines ⇒ 2 - 1 3 6 3 1 α 2 - 2 = 0 ⇒ 2 - 6 - 2 + 1 - 12 - α + 3 12 - 3 α = 0 ⇒ - 16 - 12 - α + 36 - 9 α = 0 ⇒ - 10 α + 8 = 0 ⇒ α = 4 5 Case- 2 : Parallel lines - α 6 = - 2 3 , - α 2 = - 2 - 1 ⇒ α = 4 , α = - 4 ⇒ P = 16 + 16 + 16 25 [ P ] = 32 + 16 25 = 32