Tangents to a Parabola From Latus Rectum Meet at Right Angles
\[y^2=2px\]
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. 
