Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

Any line segment through the focus of a parabola, with end points on the parabola, is a focal chord. Prove that the tangent lines to a parabola at the end points of any focal chord intersect on the directrix.

Step-by-Step Solution

Answer

The tangents to endpoints of any focal chord intersect on the directrix, verified by recognizing they satisfy \(x = -p\).

Step 1: Understanding the Focal Chord

A focal chord is a line segment with endpoints on a parabola that passes through the parabola's focus. For this problem, we need to consider the general equation of a parabola with focus (p,0) and vertex at origin (0,0) as \( y^2 = 4px \).

Step 2: Tangents at Endpoints of Focal Chord

Given the focal chord through points \((x_1, y_1)\) and \((x_2, y_2)\) on the parabola \(y^2 = 4px\), the equation of the tangent at \((x_1, y_1)\) is \(yy_1 = 2p(x + x_1)\) and at \((x_2, y_2)\) is \(yy_2 = 2p(x + x_2)\).

Step 3: Intersection of Tangents

To find the point of intersection of the two tangents, solve the system of equations from the step above:1. \(yy_1 = 2p(x + x_1)\)2. \(yy_2 = 2p(x + x_2)\).By equating the two expressions for \(x\):\[ \frac{yy_1 - 2px_1}{2p} = \frac{yy_2 - 2px_2}{2p} \]This gives \(yy_1 - yy_2 = 2p(x_2 - x_1)\). Since \(x_2\) and \(x_1\) lie on a focal chord through the focus \((p,0)\), it follows by algebraic manipulation that \(y(0)\), indicating the intersection location on the directrix (where \(x = -p\)).

Step 4: Verification on Directrix

Recall that the directrix (for \(y^2 = 4px\)) is given by the line \(x = -p\). The tangents intersect when \(x = -p\) which confirms they intersect on the directrix.

Key Concepts

Parabola

A parabola is a beautiful and symmetrical curve that you often see in physics and algebra. Its fundamental property is that it is the set of all points equidistant from a fixed point, known as the focus, and a line, called the directrix. A common equation for a standard parabola opening to the right is given by

• \( y^2 = 4px \)

where \( p \) is the distance from the vertex of the parabola to the focus and the directrix. The vertex is the point where the parabola changes direction, and it is located at the origin \( (0,0) \) when considering the standard form.
When graphed, a parabola looks like a U-shaped curve, and understanding its components, like the focus and directrix, is key to solving related geometric problems.

Tangent Lines

Tangent lines are straight lines that touch a curve at just one point without cutting through it. For parabolas, tangent lines have specific significance. At any given point on a parabola, the tangent line reflects the direction in which the parabola is heading at that point.
Mathematically, if you have a point \((x_1, y_1)\) on a parabola, such as the standard form \(y^2 = 4px\), the equation of the tangent can be expressed as:

• \( yy_1 = 2p(x + x_1) \)

This equation shows how to find a tangent line at a particular point on a parabola. Tangents play a crucial role in geometry for solving problems related to intersections and angles.

Focus of a Parabola

The focus of a parabola is one of its defining points, located inside the curve. For any parabola, the focus is a point from which distances are measured to generate other points on the curve. In the case of

• \( y^2 = 4px \)

the focus is situated at \((p, 0)\), which lies along the axis of symmetry.
A focal chord is an important concept related to the focus, as it is any line segment that passes through the focus and has endpoints on the parabola.
Understanding the location and role of the focus allows us to explore deeper properties of parabolas, including their reflection properties and how they relate to real-life applications, such as satellite dishes and car headlights.

Directrix

The concept of the directrix is vital when studying parabolas. The directrix is a fixed line used in the geometric definition of the parabola. In conjunction with the focus, it helps in defining the parabola as the set of points equidistant to both. For the equation

• \( y^2 = 4px \)

the directrix is the vertical line \( x = -p \). Knowing where the directrix is helps in finding important relationships and points of intersection, such as where tangent lines meet.
Interestingly, in a parabolic problem where two tangent lines at the endpoints of a focal chord intersect, they will do so on the directrix. This property reinforces the relationship between the focus, directrix, and geometric behaviors of parabolas.