## Tangents to a Parabola From Latus Rectum Meet at Right Angles

The latus rectum of the parabola
$y^2=2px$
is the vertical line passing through the focal point
$( \frac{p}{2}, 0)$
of the parabola, perpendicular to the
$x$
axis.
The latus rectum meets the parabola at the point
$( \frac{p}{2}, \sqrt{2p \frac{p}{2}} )=( \frac{p}{2}, \pm p)$
.
The gradient function of the parabola is
$2y \frac{dy}{dx} =2p \rightarrow \frac{dy}{dx}=\frac{p}{y}$
.
The tangents are given by
$y- \pm p= \frac{p}{ \pm p}(x- \frac{p}{2})$
which can be rearranged to give
$y= x+ \frac{p}{2}), \; y= - x - \frac{p}{2}$
.
The tangents meet at
$(- \frac{p}{2} , 0)$
at have gradients 1 and -1 respectively, so are perpendicular.