Let a , b , c be the length of three sides of a triangle satisfying the condition a 2 + b 2 x 2 − 2 b a + c x + b 2 + c 2 = 0 . If the set of all possible values of x is in the interval α , β , then 12 α 2 + β 2 is equal to _______.

Step-by-step solution

Given, a 2 + b 2 x 2 − 2 b a + c x + b 2 + c 2 = 0 ⇒ a 2 x 2 − 2 a b x + b 2 + b 2 x 2 − 2 b c x + c 2 = 0 ⇒ a x − b 2 + b x − c 2 = 0 ⇒ a x − b = 0 , b x − c = 0 ⇒ a x = b , b x = c Now, we know that the sum of two sides is always greater than the third side of a triangle, Now, taking a + b > c we get, ⇒ a + a x > b x ⇒ a + a x > a x 2 ⇒ x 2 − x − 1 < 0 ⇒ 1 - 5 2 < x < 1 + 5 2 . . . i Similarly, for b + c > a we get, ⇒ x 2 + x - 1 > 0 ⇒ x ∈ - ∞ , − 1 − 5 2 ∪ − 1 + 5 2 , ∞ . . . . i i And for c + a > b we get, ⇒ x