If the line y = m x bisects the area enclosed by the lines x = 0 , y = 0 , x = 3 2 and the curve y = 1 + 4 x - x 2 , then 12 m is equal to .

Step-by-step solution

Given, the equation of line, y = m x And the line x = 0 ,   y = 0 ,   x = 3 2   The intersection point 3 2 , 0 Area of triangle is half the total area 1 2 ∫ 0 3 / 2 1 + 4 x - x 2 d x = ∫ 0 3 / 2 m x d x 1 2 x + 2 x 2 - x 3 3 0 3 / 2 = m 2 x 2 0 3 / 2 3 2 + 9 2 - 9 8 = 9 m 4 39 8 = 9 m 4 m = 39 18 Then, 12 m 12   m = 39 18 × 12 12 m = 26