If the points A : 0 , a , B : - 2 , 0 and C : 1 , 1 form an obtuse angle triangle (obtuse angled at angle A ), then sum of all the possible integral values of a is

Step-by-step solution

Let, the point 0 , a lie inside the circle whose end-points of the diameter are - 2,0 and 1,1 , but does not lie on this diameter ⇒ 0 , a lie inside the circle x + 2 x - 1 + y y - 1 = 0 ⇒ 0 + 2 0 - 1 + a a - 1 < 0 ⇒ a 2 - a - 2 < 0 ⇒ a - 2 a + 1 < 0 ⇒ a ∈ - 1 , 2 But, a ≠ 2 3 because at a = 2 3 , points 0 , 2 3 , - 2 , 0 and 1 , 1 are collinear ⇒ a ∈ - 1 , 2 - 2 3 Hence, the sum of all possible integral values of ‘ a ’ is 1