Tangents to a Parabola From Latus Rectum Meet at Right Angles

The latus rectum of the parabola  
  is the vertical line passing through the focal point  
\[( \frac{p}{2}, 0)\]
  of the parabola, perpendicular to the  
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.

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