Math CentralQuandaries & Queries


Question from Marcus, a student:

Show that the tangent lines to the parabola y = ax^2 + bx + c at any two points with x-coordinates p and q must intersect at a point whose x-coordinate is halfway between p and q.


Suppose the two points are P and Q and have coordinates P(p, ap2 + bp + c) and Q(q, aq2 + bq + c). Use the derivative to find the slope of the tangent lines at P and Q and then write the equations of the two tangent lines, one at P and the other at Q. Set the y-coordinates of the two equations equal to each other and solve for x. You will find the x = (p + q)/2.


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