Answer
The tangent lines intersect at a point whose x-coordinate is halfway between $p$ and $q$.
Work Step by Step
$y = ax^2+bx+c$
$y' = 2ax+b$
Suppose there is a tangent line at the x-coordinate $x=p$. We can find the equation of the tangent line:
$y-(ap^2+bp+c) = (2ap+b)(x-p)$
$y = 2apx-ap^2+bx+c$
Suppose there is a tangent line at the x-coordinate $x=q$. We can find the equation of the tangent line:
$y-(aq^2+bq+c) = (2aq+b)(x-q)$
$y = 2aqx-aq^2+bx+c$
We can find the x-coordinate of the point where these two tangent lines intersect:
$y = 2apx-ap^2+bx+c = 2aqx-aq^2+bx+c$
$2px-p^2 = 2qx-q^2$
$2px-2qx = p^2-q^2$
$x~(2p-2q) = p^2-q^2$
$x = \frac{p^2-q^2}{2p-2q}$
$x = \frac{(p-q)(p+q)}{2(p-q)}$
$x = \frac{p+q}{2}$
The point $\frac{p+q}{2}$ is halfway between $p$ and $q$.
Therefore, the tangent lines intersect at a point whose x-coordinate is halfway between $p$ and $q$.
