Problem 81 - Thomas Calculus

Question

The reflective property of parabolas The accompanying figure shows a typical point \(P\left(x_{0}, y_{0}\right)\) on the parabola \(y^{2}=4 p x\) . The line \(L\) is tangent to the parabola at \(P .\) The parabola's focus lies at \(F(p, 0) .\) The ray \(L^{\prime}\) extending from \(P\) to the right is parallel to the \(x\) -axis. We show that light from \(F\) to \(P\) will be reflected out along \(L^{\prime}\) by showing that \(\beta\) equals \(\alpha .\) Establish this equality by taking the following steps. $$ \begin{array}{l}{\text { a. Show that } \tan \beta=2 p / y_{0}} \\ {\text { b. Show that } \tan \phi=y_{0} /\left(x_{0}-p\right)} \\ {\text { c. Use the identity }}\end{array} $$ $$ \tan \alpha=\frac{\tan \phi-\tan \beta}{1+\tan \phi \tan \beta} $$ \(\begin{aligned} & \text { to show that } \tan \alpha=2 p / y_{0} \\ \text { since } \alpha \text { and } \beta & \text { are both acute, tan } \beta=\tan \alpha \text { implies } \beta=\alpha \end{aligned}\) $$ \begin{array}{l}{\text { This reflective property of parabolas is used in applications like }} \\ {\text { car headlights, radio telescopes, and satellite TV dishes. }}\end{array} $$

Step-by-Step Solution

Answer

Light from the focus reflects off the parabola so that \(\alpha = \beta\).

Step 1: Find the Slope of the Tangent Line to the Parabola

Consider the parabola given by the equation \(y^2 = 4px\). To find the slope of the tangent at a point \((x_0, y_0)\) on the parabola, use implicit differentiation. Differentiate both sides of \(y^2 = 4px\) with respect to \(x\):\[ 2y \frac{dy}{dx} = 4p \]Rearranging, we find \(\frac{dy}{dx} = \frac{2p}{y}\). Since \((x_0, y_0)\) lies on the parabola, replace \(y\) with \(y_0\), giving \(\frac{dy}{dx} = \frac{2p}{y_0}\). Hence, \(\tan \beta = \frac{2p}{y_0}\).

Step 2: Calculate the Slope of Line FP

The point \(F\) is the focus \((p, 0)\) and \(P\) is \((x_0, y_0)\). The slope of line FP, which makes angle \(\phi\) with the x-axis, is given by:\[ \tan \phi = \frac{y_0 - 0}{x_0 - p} = \frac{y_0}{x_0 - p} \].

Step 3: Use the Angle Difference Identity to Find \(\tan \alpha\)

Given the identity:\[ \tan \alpha = \frac{\tan \phi - \tan \beta}{1 + \tan \phi \tan \beta} \]substitute \(\tan \phi = \frac{y_0}{x_0 - p}\) and \(\tan \beta = \frac{2p}{y_0}\) into the identity:\[ \tan \alpha = \frac{\frac{y_0}{x_0 - p} - \frac{2p}{y_0}}{1 + \frac{y_0}{x_0 - p} \cdot \frac{2p}{y_0}} \]Simplify the expression to attain that \(\tan \alpha = \frac{2p}{y_0}\), which indicates that \(\beta = \alpha\) since both \(\alpha\) and \(\beta\) are acute angles.

Key Concepts

Parabolas

Parabolas are fascinating curves with various unique properties, often seen in fields like physics, engineering, and computer graphics. A parabola is the set of all points in a plane that are equidistant from a fixed point called the "focus" and a fixed line known as the "directrix." The general equation of a parabola with its vertex at the origin and opening to the right is given by\[y^2 = 4px\]where \(p\) is the distance from the vertex to the focus. Understanding this equation helps us analyze attributes like the vertex, axis of symmetry, and its curvature. In many applications, parabolas are leveraged for their reflective properties, such as in satellite dishes and car headlights. Recognizing how light or waves are focused or unfocused due to a parabola's shape is key in these designs.

Tangent Line

A tangent line is a line that touches a curve at exactly one point and has the same slope as the curve at that point. For a parabola defined by the equation \(y^2 = 4px\), the tangent line at a point \((x_0, y_0)\) on the curve will have a unique slope. This slope is crucial because it tells us how steep the curve is at that point.
To find this slope for a parabola, we can use implicit differentiation. This involves differentiating both sides of the equation with respect to \(x\), yielding the slope \(\frac{dy}{dx}\) as\[\frac{dy}{dx} = \frac{2p}{y_0}\]This expression shows how rapidly the parabola's curve is changing at the specific point \((x_0, y_0)\). This tangent line plays an essential role in determining the reflective properties of the parabola, particularly in the context of the reflective property exercise.

Implicit Differentiation

Implicit differentiation is a technique used when we have an implicit equation rather than an explicit one. It's particularly useful when solving problems involving curves that are not easily solvable for one variable.
In the context of a parabola \(y^2 = 4px\), both \(y\) and \(x\) are entangled in the equation. We differentiate both sides with respect to \(x\) to find the derivative \(\frac{dy}{dx}\). This process allows us to assess the slope of the tangent line at any given point. Differentiating the given equation \[2y \frac{dy}{dx} = 4p\]and solving for \(\frac{dy}{dx}\) allows us to expressthe rate at which \(y\) changes with respect to \(x\) at any point as \[\frac{dy}{dx} = \frac{2p}{y_0}\]This value is crucial when analyzing the geometric properties of parabolas, especially when understanding how they can reflect light or sound.

Focus of a Parabola

The focus of a parabola is a vital point that significantly influences the parabola's geometric and reflective properties. For a parabola with the equation \(y^2 = 4px\), the focus is located at the point \((p, 0)\). This point is a fixed position that, along with the directrix, helps to define the parabola's shape. The focus is important because it is the point from which incoming parallel lines (like light rays) will reflect off the parabola and pass through a common point, or vice versa.
This unique reflective property means that a ray emanating from the focus of the parabola will be reflected outward as a parallel ray to the axis of symmetry. This principle is fundamental in designing parabolic reflectors and antennas, as it ensures effective concentration or dispersion of waves. Understanding the role of the focus in such designs highlights the intersection of geometry and practical engineering applications.