A line has equation y = kx + 6 and a curve has equation y = x^2 + 3x + 2k, where k is a constant.
(i) For the case where k = 2, the line and the curve intersect at points A and B. Find the distance AB and the coordinates of the mid-point of AB.
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`y = kx+6`
`y = x^2+3x+2k`
when k= 2
`y = 2x+6`
At intersecting points;
`2x+6 = x^2+3x+4`
`x=-2` and `x=1`
When x= -2 then y = 2
When x = 1 then y = 8
Distance AB `= sqrt((-2-1)^2+(2-8)^2) = sqrt(3^2+6^2) = 3 sqrt(5)`
Mid point of AB `= (1/2)(-2+1),(1/2)(2+8) = (-1/2 , 5)`
So the distance AB is `3sqrt(5)` units and mid point of AB is `(-1/2,5)`
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