There is a line through the origin that divides the region bounded by the parabola y = 4 x - 2 x^2 and the x-axis into two regions with....?

設此直線為 y = mx
並設此直線與題目的拋物線交點為 ( a , ma )
繪出拋物線的簡圖可知:
(1) 拋物線與 x 軸交於 x = 0 與 x = 2
(2) 拋物線與 x 軸所圍成的區域在第一象限,
因此: m > 0 , 0 < a < 2

交點 ( a , ma ) 通過 y = mx 與 y = 4x - 2x²
因此 :
ma = 4a - 2a²
2a² + ( m - 4 )a = 0
a( 2a + m - 4 ) = 0
2a + m - 4 = 0 , 因為 a > 0
a = - ( m - 4 )/2 = 2 - m/2

拋物線與 x 軸圍成的區域面積
= ∫ ( 4x - 2x² ) dx , from x = 0 to x = 2
= [ 2x² - (2/3)x³ ] , from x = 0 to x = 2
= 8 - 16/3
= 8/3

拋物線與 y = mx 圍成的區域面積
= ∫ ( 4x - 2x² - mx ) dx , from x = 0 to x = a
= [ 2x² - (2/3)x³ - (m/2)x² ] , from x = 0 to x = a
= 2a² - (2/3)a³ - (m/2)a²

2a² - (2/3)a³ - (m/2)a² = (1/2)(8/3) = 4/3
( 2 - m/2 )a² - (2/3)a³ = 4/3
a( a² ) - (2/3)a³ = 4/3
(1/3)a³ = 4/3
a³ = 4
2 - m/2 = a = 4^(1/3) = 2^(2/3)
4 - m = 2*2^(2/3) = 2^(5/3)
m = 4 - 2^(5/3)

Ans: 此直線斜率為 4 - 2^(5/3)