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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(1-21/3)
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