Problem 29 - Algebra and Trigonometry

Question

Show by eliminating the parameter \(\theta\) that the following parametric equations represent a hyperbola: $$ x=a \tan \theta \quad y=b \sec \theta $$

Step-by-Step Solution

Answer

The parametric equations represent a hyperbola: \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \).

Step 1: Express Tan in terms of x

From the first parametric equation, we have \( x = a \tan \theta \). We can express \( \tan \theta \) in terms of \( x \) by rearranging this equation: \( \tan \theta = \frac{x}{a} \).

Step 2: Express Sec in terms of y

From the second parametric equation, we have \( y = b \sec \theta \). Rearrange this to solve for \( \sec \theta \): \( \sec \theta = \frac{y}{b} \).

Step 3: Use the Pythagorean Identity

Recall the Pythagorean identity for tangent and secant: \( \sec^2 \theta - \tan^2 \theta = 1 \). Substitute the expressions from Steps 1 and 2 into this identity: \( \left(\frac{y}{b}\right)^2 - \left(\frac{x}{a}\right)^2 = 1 \).

Step 4: Rearrange to Form the Equation of a Hyperbola

Plug in the expressions for \( \sec \theta \) and \( \tan \theta \) into the identity: \[ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \]. This is the standard form of the equation of a hyperbola.

Key Concepts

Understanding Hyperbolas

Hyperbolas are types of conic sections that look like two mirrored curves. They occur naturally in many real-world applications, especially in optical systems and navigation.

• Standard Form: A hyperbola’s standard equation is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \).
• Axes: A hyperbola has two axes—the transverse axis, where the two branches stretch away from each other, and the conjugate axis, which is perpendicular to the transverse axis.
• Foci: Each branch of the hyperbola twists away from a central point, known as the center, and appears toward the foci, which are points located symmetrically along the transverse axis.
• Vertices: The closest points of each branch to the center of a hyperbola are called vertices.

Understanding these parts can make it easier to recognize and plot hyperbolas from their equations. In our case, by altering the parametric equations, we obtained such a relationship—in line with the above-mentioned standard equation of a hyperbola.

Pythagorean Identity Relevance

The Pythagorean identity becomes relevant when dealing with trigonometric functions in parametric equations. It's especially useful because it connects tangent and secant in a powerful way.
It states:

• \( \sec^2 \theta - \tan^2 \theta = 1 \)

This identity provides a smooth pathway to move from trigonometric functions to algebraic equations. In our exercise, we utilize this identity to eliminate the parameter \( \theta \) and make connections between \( x \) and \( y \).
The identity helps us reformulate the parametric conditions into a recognizable algebraic form, which is precisely why it’s pivotal for converting parametric equations into the equation of a hyperbola.

Role of Tangent in Parametric Equations

The tangent function is key when dealing with hyperbolic structures because it relates an angle to a ratio, perfectly aligning with the concepts needed in conic sections.

• Definition: In a right triangle, the tangent of an angle \( \theta \) is the ratio of the opposite side to the adjacent side.
• In our equation: From the parametric equation \( x = a \tan \theta \), tangent helps express \( x \) in terms of the angle \( \theta \).

This parameterization makes it possible to transform complexities associated with angles into straightforward values, facilitating the manipulation of equations.
By expressing tangent as \( \tan \theta = \frac{x}{a} \), we mold our equation into forms conducive to further algebraic manipulation, essential when aimed at deriving hyperbolic conditions.

Understanding the Secant Function

Secant, frequently less understood than its trigonometric counterparts, plays an integral role in transforming angular data into algebraic expressions.

• Definition: Secant of an angle \( \theta \) in a right triangle is the reciprocal of the cosine of \( \theta \), expressed as \( \sec \theta = \frac{1}{\cos \theta} \).
• In application: As shown in our parametric equation \( y = b \sec \theta \), we express \( y \) in terms of \( \theta \).

Enabling expressions for \( \sec \theta \) such as \( \sec \theta = \frac{y}{b} \), secant aids in bridging gaps between angular and algebraic expressions.
This transition fosters easier handling of parametric forms, critically needed to simplify solutions into recognizable hyperbolas, palpable through our found solution \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \).