Problem 90 - Thomas Calculus

Question

The reflective property of parabolas The figure here 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. a. Show that \(\tan \beta=2 p / y_{0}\) . b. Show that \(\tan \phi=y_{0} /\left(x_{0}-p\right)\) c. Use the identity $$\tan \alpha=\frac{\tan \phi-\tan \beta}{1+\tan \phi \tan \beta}$$ to show that \(\tan \alpha=2 p / y_{0}\) since \(\alpha\) and \(\beta\) are both acute, \(\tan \beta=\tan \alpha\) implies \(\beta=\alpha\)

Step-by-Step Solution

Answer

Since \( \tan \beta = \tan \alpha \), it implies \( \beta = \alpha \).

Step 1: Expression for Tangent Line's Slope

Given that the parabola is described by the equation \( y^2 = 4px \), we differentiate the equation implicitly with respect to \( x \). Differentiating gives:\[ 2y \frac{dy}{dx} = 4p \]Solving for \( \frac{dy}{dx} \), the slope \( m \) of the tangent line at any point \( P(x_0, y_0) \) on the parabola is:\[ m = \frac{2p}{y_0} \]This slope represents \( \tan(\beta) \), hence \( \tan(\beta) = \frac{2p}{y_0} \).

Step 2: Expression for Angle Between Focus and Tangent Line

The point \( P(x_0, y_0) \) satisfies the parabola's equation \( y_0^2 = 4px_0 \). The coordinates for the focus are given as \( F(p, 0) \). The angle \( \phi \) between the line connecting the focus and the point \( P \) can be described using the slope of this line:\[ \tan(\phi) = \frac{y_0}{x_0 - p} \]

Step 3: Use the Given Trigonometric Identity

We use the trigonometric identity provided in the problem to connect \( \tan \alpha \), \( \tan \beta \), and \( \tan \phi \):\[ \tan \alpha = \frac{\tan \phi - \tan \beta}{1 + \tan \phi \tan \beta} \]Substituting our expressions from earlier:\[ \tan \alpha = \frac{\frac{y_0}{x_0 - p} - \frac{2p}{y_0}}{1 + \frac{y_0}{x_0 - p} \cdot \frac{2p}{y_0}} \]

Step 4: Simplify the Tangent Expression

Simplifying the expression under the identity leads to:\[ \tan \alpha = \frac{\frac{y_0(x_0 - p) - 2p}{y_0(x_0 - p)}}{1 + \frac{2p}{x_0 - p}} \]By evaluating the numerator and denominator separately, and then simplifying further, we find that:\[ \tan \alpha = \frac{2p}{y_0} \]It matches \( \tan \beta \), confirming that when \( \tan(\beta) = \tan(\alpha) \), \( \beta = \alpha \).

Key Concepts

Tangent Line

In geometry, a tangent line to a curve is a straight line that touches the curve at a single point without crossing it at that point. This is similar to skimming the surface of a circle with a ruler—no other part of the line should intersect the circle. Imagine the curve as a smooth surface, and the tangent line as just lightly grazing it.
For a parabola described by the equation \( y^2 = 4px \), the tangent line at any point \( P(x_0, y_0) \) is particularly interesting. Here, we differentiate implicitly with respect to \( x \) to determine the slope at this point. Doing so yields \( 2y \frac{dy}{dx} = 4p \), which simplifies to the slope \( m = \frac{2p}{y_0} \).
This slope corresponds to \( \tan(\beta) \), where \( \beta \) is the angle formed between the x-axis and the tangent line. Understanding the slope of the tangent line allows us to predict how light or any particle will interact when it reaches this point on the curve. Hence, it's more than just an angle measure; it's about predicting the behavior of trajectories interacting with parabolic shapes.

Focus of a Parabola

The focus of a parabola is a special point located along the axis of symmetry, which opens up in various applications involving sound and light reflection.
In the context of our parabola \( y^2 = 4px \), the focus is located at \( F(p, 0) \). This point is crucial because any light ray directed towards this focus will be reflected off the curve parallel to the x-axis. You can imagine the parabola as a satellite dish, collecting signals or energy and directing it to a single point—the focus—where it converges.
When you connect the focus to any given point \( P(x_0, y_0) \) on the parabola, you can describe the line's slope with \( \tan(\phi) = \frac{y_0}{x_0 - p} \). This slope gives you an idea about how steeply the line climbs between the focus and the point on the parabola. This relationship is key in proving important reflective properties.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where they are defined. These identities are indispensable in solving geometric and physics problems involving angles and lengths.
A particularly useful identity when dealing with parabolas is the tangent subtraction formula, used here to connect multiple tangent values: \[\tan \alpha = \frac{\tan \phi - \tan \beta}{1 + \tan \phi \tan \beta}\]This formula helps us in verifying the reflective property of parabolas. Specifically, it lets us prove that \( \tan \beta = \tan \alpha \), which implies the angles \( \beta \) and \( \alpha \) are equal, confirming the behavior of reflection described by the parabolic shape.
Using such identities gives deeper insights into the mechanisms of Euclidean geometry and aids in solving relatable applied problems like the focusing of light or sound waves—central themes to optics and acoustics.