Question

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.

Short answer

Answer: For any point P on the given parabola to the left of the latus rectum, the sum of the shortest distance from P to the latus rectum and the distance from P to the focus is a constant value of p.

Step-by-step solution

Find the equation of the latus rectum and the coordinates of the focus

: The given equation of the parabola is \(y^2=4px\). The focus of the parabola is at \((p,0)\), and the latus rectum passes through the focus and is perpendicular to the major axis (\(x\)-axis) of the parabola. The equation of the latus rectum can be written as \(x=p\).

Find the distance between point P and the latus rectum

: Let \(P\) be any point on the parabola to the left of the latus rectum. Since \(P\) lies on the parabola, its coordinates can be written as \((x_P,y_P)\), where \(y_P^2 = 4px_P\). The distance \(D\) between point \(P\) and the latus rectum (line \(x=p\)) can be found using the formula for the distance between a point and a vertical line: \(D = |x_P - p|\).

Find the distance between point P and the focus

: Let \(F\) be the focus of the parabola with coordinates \((p,0)\). The distance between point \(P\) and the focus \(F\) can be found using the distance formula: \(|FP| = \sqrt{(x_P - p)^2 + (y_P - 0)^2}\)

Show that the sum of the distances D and |FP| is constant

: Next, we need to show that the sum of the distances \(D\) and \(|FP|\) is constant for all points \(P\) on the curve. We can do this by adding the expressions for these distances and showing that the result is independent of the coordinates of \(P\): $$ D + |FP| = |x_P - p| + \sqrt{(x_P - p)^2 + y_P^2} $$ Substitute \(y_P^2 = 4px_P\): $$ D + |FP| = |x_P - p| + \sqrt{(x_P - p)^2 + 4px_P} $$ If we can show that this expression is equal to a constant value for all points on the parabola, we will have proven the result.

Find the constant value

: Now consider the vertex of the parabola, \(V\), which is located at \((0,0)\). We can find the value of the expression \(D + |FP|\) at this point: $$ D_{V} + |FV| = |0 - p| + \sqrt{(0 - p)^2 + 4p(0)} = p $$ The expression \(D_{V} + |FV| = p\) is a constant for this parabola, so we have proven that for any point \(P\) on the parabola to the left of the latus rectum, the sum of the shortest distance from \(P\) to the latus rectum and the distance from \(P\) to the focus is a constant value of \(p\).

Key concepts

Conic Sections and Parabolas

In mathematics, conic sections are the curves obtained by intersecting a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas. Among these, the parabola is unique in that any point on its curve is equidistant from the focus (a fixed point) and the directrix (a fixed straight line).

When learning about parabolas, understanding the standard form of its equation is crucial. In the case of a parabola that opens horizontally, like the one in our exercise, the equation is given as \(y^2 = 4px\), where \(p\) is the distance from the vertex to the focus and also from the vertex to the directrix.

The latus rectum of a parabola is a special focal chord, as it is perpendicular to the axis of symmetry of the parabola and passes through the focus. It plays an integral role in understanding the geometric properties of parabolas.

Focal Chord and Latus Rectum

The concept of a focal chord is tied directly to the unique properties of a conic section's focus. A focal chord is any line segment that passes through the focus of a conic section and has both endpoints on the curve itself. In a parabola, all focal chords exhibit interesting properties relevant to their position and length.

In a parabola with equation \(y^2 = 4px\), the latus rectum is a focal chord that is horizontal and thus perpendicular to the axis of symmetry. Its significance comes from the fact that the endpoints of the latus rectum lie equidistant from the vertex and have the same 'y' coordinate, thereby simplifying many geometric and algebraic calculations involving the parabola. The length of the latus rectum is especially notable because it is directly proportional to the parameter \(p\), which itself is a measure of how 'wide' or 'narrow' the parabola is.

Distance Formula

A fundamental tool in geometry is the distance formula, which is used to calculate the distance between two points in the Cartesian coordinate system. The formula is derived from the Pythagorean theorem and is given by:\[|AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

This formula is important when studying conic sections because it allows us to determine the length of line segments such as focal chords. For the example of the parabola \(y^2 = 4px\), we use the distance formula to calculate the distances between any point \(P\) on the parabola and the latus rectum, as well as between point \(P\) and the focus \(F\). The beauty of the distance formula in the context of conic sections is that it provides a bridge between algebraic expressions and geometric interpretations, facilitating a deeper understanding of the structures in question.