Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

Let L be the latus rectum of the parabola \(y^{2}=4 p x\) for \(p>0\). Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D+|F P| is a constant. Find the constant.

Solution 91AE

Step 1:

A Focal chords of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic.

Prove the following properties.

Let L be the latus rectum of the parabola for p > 0. Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D + ∣FP∣ is a constant. Find the constant.