If ( a , b ) be the orthocentre of the triangle whose vertices are ( 1 , 2 ) , ( 2 , 3 ) and ( 3 , 1 ) , and I 1 = ∫ a b xsin 4 x - x 2 dx , I 2 = ∫ a b sin 4 x - x 2 dx , then 36 I 1 I 2 is equal to :

Step-by-step solution

Equation of CE is given by, ⇒ y - 1 = - 1 3 - 2 2 - 1 x - 3 ⇒ y - 1 = - ( x - 3 ) ⇒ x + y = 4 . . . i Similarly, equation of A D is given by, ⇒ y - 2 = - 1 3 - 1 2 - 3 x - 1 ⇒ y - 2 = 1 2 x - 1 ⇒ 2 y - 4 = x - 1 ⇒ x - 2 y + 3 = 0 Using i ⇒ 4 - y - 2 y + 3 = 0 ⇒ y = 7 3 ⇒ x = 4 - 7 3 ⇒ x = 5 3 So, Orthocentre is H ≡ a , b = 5 3 , 7 3 . Orthocentre lies on the line x + y = 4 So, a + b = 4 Now, I 1 = ∫ a b x sin ( x ( 4 - x ) ) d x . . . ( 1 ) ⇒ I 1 = ∫ a b ( a + b - x ) sin ( a + b - x ( 4 - a + b - x ) ) d x ⇒ I 1 =