Hi,
How do you prove that the projection of the intersection point between two tangents of the parabola projected to the x-axix is at the midpoint between the two points (of tangents with the parabola) on the x axis.?
Thanks
Hi,
Let P be the parabola $\displaystyle ax^2 + bx + c$
One Cartesian equation of the tangent of the parabola at point $\displaystyle x_0, y_0$ is
$\displaystyle y = (2a x_0 + b)x - a {x_0}^2 + c$
The abscissa of the intersection point between two tangents of the parabola at points $\displaystyle x_0, y_0$ and $\displaystyle x_1, y_1$ is given by
$\displaystyle (2a x_0 + b)x - a {x_0}^2 + c = (2a x_1 + b)x - a {x_1}^2 + c $
Simplification leads to
$\displaystyle 2a (x_0 - x_1)x - a (x_0 - x_1)(x_0 + x_1) = 0$
a being non-zero and x0 being different from x1
$\displaystyle x = \frac{x_0 + x_1}{2}$