Math CentralQuandaries & Queries


Question from James, a student:

There is a line through the origin that divides the region bounded by the parabola y=3x-5x^2 and the x-axis into two regions with equal area. What is the slope of that line?

Hi James,

y = 3x - 5x2 = x(3 - 5x) crosses the x-axis at x = 0 and x = 5/3 so find the area of the region bounded by y = 3x - 5x2 and the x-axis by integrating 3x - 5x2 from x = 0 to x = 5/3. Call the value A.

Suppose the line through the prigin y = mx intersects the parabola at Q with x-coordinate p.

parabola and line

The task is to find m so that


Evaluation of the integral gave me

1/2 (3 -m) p2 - 5/3 p3 = A/2                             (1)

The relationship between p and m comes from the fact that the point Q is on the line y = mx and also on the parabola y = 3x - 5x2. Thus mp = 3p - 5p2 or p = (3 - m)/5. Substitute into (1) to get

1/2 (3 - m) (3 - m)2/25 -5/3 (3 - m)3/125 = A/2

Solve for m.


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