Three points O 0 , 0 , P a , a 2 , Q − b , b 2 , a > 0 , b > 0 , are on the parabola y = x 2 . Let S 1 be the area of the region bounded by the line P Q and the parabola, and S 2 be the area of the triangle O P Q . If the minimum value of S 1 S 2 is m n , gcd m , n = 1 , then m + n is equal to:

Step-by-step solution

Plotting the diagram of the given data we get, Area of △ O P Q is given by, S 2 = 1 2 0 0 1 a a 2 1 − b b 2 1 ⇒ S 2 = 1 2 a b 2 + a 2 b . . . i Now, equation of line P Q : y − a 2 = a 2 − b 2 a + b x − a ⇒ y − a 2 = a − b x − a − b a ⇒ y = a − b x + a b . . . i i Area bounded by the line P Q and the parabola is given by, S 1 = ∫ − b a a − b x + a b − x 2 d x ⇒ S 1 = a − b x 2 2 + a b x − x 3 3 − b a ⇒ S 1 = a − b 2 a + b 2 + a b a + b − a 3 + b 3 3 . . . i i i ⇒ S 1 S 2 = a − b 2 2 + a b − a 2 + b 2 − a b 3 a b 2 ⇒