The curve xy = 11x + 5 cuts the line y = x + 10 at the points A and B. The mid-point of AB is the point C. Show that the point C lies on the line x + y = 11.

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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

* 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.

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