# 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....?

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 equal area. What is the slope of that line?

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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)

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