Problem 35 - 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 define a hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1\).

Step 1: Expressing Trigonometric Functions

Start with the given parametric equations: \[x = a \tan \theta\] \[y = b \sec \theta\]Recognize that \(\sec \theta = \frac{1}{\cos \theta}\), which implies \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Step 2: Express \(\tan \theta\) in terms of \(x\)

From the equation \(x = a \tan \theta\), solve for \(\tan \theta\):\[\tan \theta = \frac{x}{a}\]

Step 3: Express \(\sec \theta\) in terms of \(y\)

From the equation \(y = b \sec \theta\), solve for \(\sec \theta\):\[\sec \theta = \frac{y}{b}\]

Step 4: Relate \(\tan \theta\) and \(\sec \theta\) using an identity

Use the trigonometric identity \(\sec^2 \theta = 1 + \tan^2 \theta\). Substitute the expressions for \(\tan \theta\) and \(\sec \theta\):\[\sec^2 \theta = 1 + \tan^2 \theta \ \left(\frac{y}{b}\right)^2 = 1 + \left(\frac{x}{a}\right)^2\]

Step 5: Simplify and Rearrange

Expand and rearrange the equation obtained:\[\frac{y^2}{b^2} = 1 + \frac{x^2}{a^2} \]Rearrange to emphasize the hyperbolic form:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1\]

Step 6: Conclusion on Hyperbola Form

The derived equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1\) represents a hyperbola equation in standard form. The parameter \(\theta\) is successfully eliminated, showing that the original parametric equations define a hyperbola.

Key Concepts

Parametric Equations

Parametric equations are a pair of equations that express the coordinates of the points on a curve as functions of a variable, often denoted as \( t \), or in this case, \( \theta \). These kinds of equations are useful for describing motion and curves where the position at each point in time is important. Rather than using a single equation for \( y \) in terms of \( x \), they define both \( x \) and \( y \) in terms of a third variable.
In the given exercise, the parametric equations \( x = a \tan \theta \) and \( y = b \sec \theta \) are used to describe a hyperbola. This means any value of \( \theta \) yields a pair \( (x, y) \) that lies on the hyperbola. This representation is helpful because it describes the curve with two distinct sets of data revolving around a parameter rather than a direct relationship between \( x \) and \( y \).
By eliminating the parameter \( \theta \), we can derive a direct relationship, which helps in understanding the shape and properties of the hyperbola itself.

Trigonometric Identities

Trigonometric identities are mathematical equations that expand the relationships between the angles and lengths in a right triangle. They are vital in simplifying expressions involving trigonometric functions. Two fundamental identities useful in this context are the definitions of tangent and secant:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• \( \sec \theta = \frac{1}{\cos \theta} \)

Additionally, the Pythagorean identity \( \sec^2 \theta = 1 + \tan^2 \theta \) plays a critical role. This relation allows us to connect \( \tan \theta \) and \( \sec \theta \) directly, facilitating the elimination of \( \theta \) in the parametric equations. Understanding these identities is crucial as they provide a stepping stone to converting parametric equations to standard forms of conic sections like hyperbolas.

Eliminating Parameters

Eliminating the parameter refers to the process of removing the extra variable \( \theta \) to find a direct connection between \( x \) and \( y \). This simplifies understanding the curve without the need for the parameter. It's like condensing a story down to its essence without extra details.
For the given equations, we start by expressing \( \tan \theta \) from \( x = a \tan \theta \):

• \( \tan \theta = \frac{x}{a} \)

Then, \( \sec \theta \) is expressed from \( y = b \sec \theta \) as:

• \( \sec \theta = \frac{y}{b} \)

Using the identity \( \sec^2 \theta = 1 + \tan^2 \theta \), substitute these expressions:

• \( \left(\frac{y}{b}\right)^2 = 1 + \left(\frac{x}{a}\right)^2 \)

Rearranging, simplifies to the equation of a hyperbola. Thus, eliminating \( \theta \) clarifies the relationship solely as one between \( x \) and \( y \).

Standard Form

The standard form of a hyperbola equation offers a straightforward way to recognize and understand its geometric properties. It looks like this:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1\]By rearranging and simplifying, this form is achieved from the derived relationship obtained by eliminating \( \theta \). The standard form is beneficial because:

• It visibly indicates the nature of the conic section as a hyperbola.
• The constants \( a \) and \( b \) provide information about the hyperbola's dimensions and orientation.

In the exercise solution, converting to standard form confirms that we have directly connected \( x \) and \( y \) through an equation characteristic of hyperbolas. It's an essential part of fully grasping the geometry of the curve and allows for further analysis such as finding asymptotes, vertices, and understanding the hyperbola's orientation.