The point a 2 , a + 1 lies in the angle between the lines 3 x + y + 1 = 0 and x + 2 y - 5 = 0 containing the origin. If a is an integer, then the sum of all possible values of a is

Step-by-step solution

a 2 , a + 1 & 0 ,0 lie on same side of line 3 x + y + 1 = 0 ⇒ 0 + 0 + 1 3 a 2 + a + 1 + 1 > 0 ⇒ 3 a 2 + a + 2 > 0 ⇒ a ∈ R … 1 a 2 , a + 1 & 0,0 also lie on same side of x + 2 y - 5 = 0 ⇒ 0 + 0 - 5 a 2 + 2 a + 2 - 5 > 0 ⇒ a 2 + 2 a - 3 < 0 ⇒ a + 3 a - 1 < 0 ⇒ a ∈ - 3,1 … 2 From 1 & 2 , we get, a ∈ - 3,1 ⇒ sum of all the integral values of ‘ a ’ is - 2 + - 1 + 0 = - 3