Question

90–94. 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 =4px 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.

Accepted Answer

Identify the key elements of the problem: the parabola is given by the equation $$y^{2} = 4px$ with p$$ \u003e 0$$, the focus F$ is at (p$$, 0)$$, and the latus rectum L$ is the vertical line through the focus, i.e., x = p$. Express the coordinates of a general point P$ on the parabola to the left of L$. Since P$ lies on y$$^{2}$$ = 4px$$, its coordinates can be written as $$P = (x, y)$$ where $$x = \frac{y$$^{2}$$}{4p}$$ and $$x \u003c p (to $$the left of $$L). $$Calculate the shortest distance $$D$$ from the point $$P to $$the latus rectum $$L. $$Since $$L is $$the vertical line $$x = p$$, the shortest distance is the horizontal distance: $$D = p - x. $$Find the distance $$|FP|$$ between the focus $$F = (p, 0)$$ and the point $$P = (x, y)$$ using the distance formula: $$|FP| = \sqrt{(x - p$$)^{2}$$ + y$$^{2}$$}. $$Set up the expression $$D + |FP|$$ and simplify it by substituting $$x = \frac{y$$^{2}$$}{4p}$$ and $$D = p - x. $$Show that this sum simplifies to a constant independent of $$P$$, and identify that constant.

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