There is a line through the origin that divides the region bounded by the parabola y = 3 x - 8 x^2 and the x-axis into two regions with equal area. What is the slope of that line?

let eq of line is y=mx            ...........1

eq of curve y = 3x-8x²           ................2

it will intersect the curve at P & Q ...

solving 1 & 2

x = 0 , (3-m)/8

y = 0 , m(3-m)/8

P = (0,0)  &  Q =[ (3-m)/8 , m(3-m)/8 ]

now x axis cuts the parabola at S = (3/8 , 0)

area bw parabola & x axis is A₁ = (3x-8x²)dx         lim 0 to 3/8

                                              =9/128 units

now area bw line & parabola is A₂ = (3x-8x² - mx)dx       lim 0 to (3-m)/8

                                               = (3-m)³/384 units

now since x axis divides the area in two equal parts so

          A₂ = 2 A₁

       (3-m)³/384 = 2(9/128)

        (3-m)³ = 27*2

         m = 3(1-2^1/3)

Approved