9514 1404 393
Explanation:
Substituting the cut line into the curve equation, we have ...
x(x+10) = 11x +5
In standard form this is ...
x^2 -x -5 = 0
The sum of the two roots of this equation is the opposite of the x-coefficient*, so is -(-1) = 1. Then the midpoint between the points of intersection has an x-coordinate of ...
(root sum)/2 = 1/2
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The midpoint y-coordinate will be the y-value on the cut line, so is ...
y = x + 10 = (1/2) +10 = 10.5
Then the midpoint of AB is C(0.5, 10.5). This is easily shown to lie on the line ...
x + y = 11
0.5 + 10.5 = 11
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* The factored form of a monic polynomial can be expanded to show this:
(x -a)(x -b) = x^2 -(a+b) +ab
That is, the sum of the roots a and b is the opposite of the x-coefficient.
