If the area enclosed by the parabolas P 1 : 2 y = 5 x 2 and P 2 : x 2 - y + 6 = 0 is equal to the area enclosed by P 1 and y = α x , α > 0 , then α 3 is equal to _____ .

Step-by-step solution

Given: P 1 : 2 y = 5 x 2 P 2 : x 2 - y + 6 = 0 On solving both equations, we have 2 y = 5 y - 6 ⇒ 3 y = 30 ⇒ y = 10 ⇒ x = ± 2 So, points of intersection are 2 , 10   &   - 2 , 10 . Also, solving P 1 and y = α x , we have x = 0 ,   2 α 5 According to question 2 ∫ 0 2 x 2 + 6 - 5 x 2 2 d x = ∫ 0 2 α 5 α x - 5 x 2 2 d x ⇒ 2 x 3 3 + 6 x - 5 x 3 6 0 2 = α x 2 2 - 5 x 3 6 0 2 α 5 ⇒ 2 12 + 8 3 - 20 3 = 2 α 3 25 - 8 α