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.

Verified step by step guidance

1. Identify the key elements of the problem: the parabola is given by the equation \(y^{2} = 4px\) with \(p > 0\), the focus \(F\) is at \((p, 0)\), and the latus rectum \(L\) is the vertical line through the focus, i.e., \(x = p\).
2. 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 \(x < p\) (to the left of \(L\)).
3. 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: \(D = p - x\).
4. Find the distance \(|FP|\) between the focus \(F = (p, 0)\) and the point \(P = (x, y)\) using the distance formula: \(|FP| = \sqrt{(x - p)^{2} + y^{2}}\).
5. 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.

Key Concepts

Parabola and Its Focus

A parabola is the set of points equidistant from a fixed point called the focus and a fixed line called the directrix. For the parabola y² = 4px (p > 0), the focus is at (p, 0). Understanding the focus is essential because focal chords and distances from points on the parabola to the focus are central to the problem.

Latus Rectum of a Parabola

The latus rectum is a special focal chord perpendicular to the axis of symmetry of the parabola. For y² = 4px, it passes through the focus and has endpoints on the parabola. Its length is 4p, and it helps define distances and geometric relationships involving points on the parabola.

Distance Between Points and Lines

Calculating the shortest distance from a point to a line involves perpendicular projection. In this problem, the distance D from a point P on the parabola to the latus rectum line is crucial. Combining this with the distance |FP| from P to the focus helps establish the constant sum property.