Problem 32 - Precalculus with Limits

Question

In Exercises 29-32, eliminate the parameter and obtain the standard form of the rectangular equation. Hyperbola: \(\quad x=h+a\ \sec\ \theta, \quad y=k + b\ \tan\ \theta\)

Step-by-Step Solution

Answer

The standard form of the given hyperbola equations is \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \)

Step 1: Express \(\sec \theta\) and \(\tan \theta\) in terms of x and y

The given equations are \(x = h + a \sec \theta\) and \(y = k + b \tan \theta\), we solve for \(\sec \theta\) and \(\tan \theta\) so that we can eliminate \(\theta\) from our equations. That gives us \(\sec \theta = \frac{x-h}{a}\) and \(\tan \theta = \frac{y-k}{b}\)

Step 2: Express \(\sec^2 \theta\) and \(\tan^2 \theta\) in terms of x and y

We know that \(\sec^2 \theta = 1+ \tan^2 \theta\) from the Pythagorean trigonometric identity. Substituting the expressions for \(\sec \theta\) and \(\tan \theta\) from step 1 gives \(\left( \frac{x-h}{a} \right)^2 = 1 + \left( \frac{y-k}{b} \right)^2\)

Step 3: Simplify the equation

Expanding and simplifying the last equation, we have \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \), which is the standard form of the equation of a hyperbola.

Key Concepts

Understanding Hyperbolas

The hyperbola is an important mathematical concept that arises in various fields such as physics and engineering. It is the set of points in a plane where the absolute difference of the distances from two fixed points (the foci) is constant. Unlike a circle or an ellipse, a hyperbola has two distinct curves, often referred to as the branches of the hyperbola.

Key features of a hyperbola include:

• Two foci and two vertices.
• Asymptotes that guide the shape of the hyperbola.
• A transverse axis that connects the vertices.

The standard form of a hyperbola's equation centered at \((h, k)\) is \[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \] for a horizontal hyperbola. This equation shows the relationship between a hyperbola and its geometric elements. The values of \(a\) and \(b\) influence the shape and orientation of the hyperbola, with \(a\) determining the distance from the center to the vertices along the transverse axis.

Exploring Parametric Equations

Parametric equations provide a flexible way to describe curves. In contrast to the standard rectangular form, which expresses \(y\) as a function of \(x\), parametric equations use a third variable, called a parameter, to express both \(x\) and \(y\).

For instance, the parametric equations for a hyperbola are given by:

• \(x = h + a \sec \theta\)
• \(y = k + b \tan \theta\)

Here, \(\theta\) serves as the parameter that varies to trace the points of the hyperbola.

Parametric equations are particularly useful because they can describe more complicated shapes and motions that rectangular equations can't easily express. These equations allow for the depiction of curves that might loop back over themselves or otherwise remain undefined in standard form, widening the scope of what can be plotted graphically.

Utilizing Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables where the functions are defined. They play a crucial role in simplifying expressions and solving equations.

One essential identity used in transforming parametric to rectangular form is the Pythagorean identity:

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

This identity helps in the process of eliminating the parameter \(\theta\) from parametric equations, simplifying them into a more familiar rectangular form. In our exercise, this identity was used to transform the equations for \(x\) and \(y\) into the standard form of a hyperbola. By expressing \(\sec \theta\) and \(\tan \theta\) in terms of \(x\) and \(y\), we could substitute and simplify into a single equation: \[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \] Hence, trigonometric identities are essential tools for manipulating and understanding complex equations.