Chapter 9.3, Problem 35WE

To determine

To prove: The length of the latus rectum is twice the distance from the focus to the directrix.

Explanation of Solution

Given information: The perpendicular line passes through the focus and intersects the parabola at two points.

Proof: Let D be the directrix line and f be the focus point.

Hence,

  $\begin{array}{l}FD=PD\\ FQ=PQ\end{array}$

As the line PQ is perpendicular to the axis: $PQ=PF+FQ$

  $\begin{array}{l}PQ=PF+FQ\\ PQ=FD+FD\\ \therefore PQ=2FD\end{array}$

The latus rectum of a parabola is the chord that passes through its focus. It is perpendicular to the major axis of a parabola. Its length is equal to the four times the focal length or just two times the distance from the focus to the directrix line.