If the normal to the curve y x = ∫ 0 x 2 t 2 - 15 t + 10 d t at a point a , b is parallel to the line x + 3 y = - 5 , a > 1 , then the value of a + 6 b is equal to ________.

Step-by-step solution

y x = ∫ 0 x 2 t 2 - 15 t + 10 d t y ' x x = a = 2 x 2 - 15 x + 10 a = 2 a 2 - 15 a + 10 Slope of normal = - 1 3 ⇒ 2 a 2 - 15 a + 10 = 3 ⇒ a = 7 and a = 1 2 (rejected) b = y 7 = ∫ 0 7 2 t 2 - 15 t + 10 d t = 2 t 3 3 - 15 t 2 2 + 10 t 0 7 ⇒ 6 b = 4 × 7 3 - 45 × 49 + 60 × 7 a + 6 b = 406