Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Show that \(x=a \sec t+h, \quad y=b \tan t+k\) \(-\pi / 2

Step-by-Step Solution

Answer

The parametric equations define a hyperbola centered at \((h, k)\), with branches for \(-\pi/2 < t < \pi/2\) and \(\pi/2 < t < 3\pi/2\).

Step 1: Understand the Parametric Equations

The given parametric equations are \(x = a \sec t + h\) and \(y = b \tan t + k\). These equations represent a curve in the \(xy\)-plane, where \(t\) is the parameter that we will modify to trace the curve.

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

From \(x = a \sec t + h\), we have \(\sec t = \frac{x-h}{a}\). From \(y = b \tan t + k\), we have \(\tan t = \frac{y-k}{b}\). These expressions link the parametric definitions to \(x\) and \(y\).

Step 3: Use Trigonometric Identity

Use the identity \(\sec^2 t - \tan^2 t = 1\). Substitute \(\sec t = \frac{x-h}{a}\) and \(\tan t = \frac{y-k}{b}\) into this identity to find the relationship between \(x\) and \(y\) for the curve.

Step 4: Derive the Hyperbola Equation

Substituting, we have \(\left(\frac{x-h}{a}\right)^2 - \left(\frac{y-k}{b}\right)^2 = 1\). Simplifying, this becomes \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), which is the standard form of the hyperbola centered at \((h, k)\).

Step 5: Identify the Axes

The derived equation \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) confirms the hyperbola's features: the transverse axis is horizontal with length \(2a\) (since the \(x\) term is positive) and the conjugate axis has length \(2b\).

Step 6: Determine Branches of the Hyperbola

Since \(\sec t\) is defined for \(-\pi/2 < t < \pi/2\) and \(\pi/2 < t < 3\pi/2\), these ranges correspond to the two branches. \(-\pi/2 < t < \pi/2\) gives one branch, and \(\pi/2 < t < 3\pi/2\) gives the other.

Key Concepts

Hyperbola

A hyperbola is an open curve formed by intersecting a plane and a double-napped cone. It consists of two symmetrical branches opening away from each other. In the context of parametric equations, the hyperbola is represented in a way that simplifies the tracing of its path using a parameter, often denoted as \(t\).
For the hyperbola equation derived from the parametric forms \(x = a \sec t + h\) and \(y = b \tan t + k\), it turns into the standard equation \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\).
The beauty of using parametric equations is that it unfolds the relationship between \(x\) and \(y\) in a stepwise fashion through parameter \(t\), making it easier to visualize how different values of \(t\) influence the curve in 2D space.

Trigonometric Identities

Trigonometric identities are essential tools in transforming and simplifying equations involving trigonometric functions. The identity \(\sec^2 t - \tan^2 t = 1\) is particularly crucial in establishing the relationship between the parametric functions and the hyperbola equation.
By replacing \(\sec t\) and \(\tan t\) with expressions derived from the parametric equations

• \(\sec t = \frac{x-h}{a}\)
• \(\tan t = \frac{y-k}{b}\)

we substitute them into the trigonometric identity:

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

This crucial step leads to the derivation of the hyperbola's equation and highlights the vital role these identities play in linking trigonometric functions to geometric shapes.

Parametric Form

Parametric form is a technique in mathematics where coordinates \((x, y)\) are expressed as functions of a parameter, \(t\). This form is highly advantageous when dealing with complex curves like hyperbolas, allowing us to break down the curve into understandable parts.
For the given exercise, the parametric equations \(x = a \sec t + h\) and \(y = b \tan t + k\) use \(t\) as a variable to dynamically explore every point on the hyperbola. These expressions allow easy manipulation and are particularly helpful in calculus for differentiation and integration, providing greater insight into the curve's properties.
Ultimately, parametric forms offer an intuitive way to understand how each part of the equation contributes to the overall shape and orientation of a hyperbola.

Hyperbola Branches

A notable feature of a hyperbola is its two distinct branches. For our parametric equations, these branches are determined by evaluating the allowable range of the parameter \(t\).
The hyperbola branches emerge due to the periodic and undefined nature of the trigonometric functions \(\sec t\) and \(\tan t\). For \(-\pi/2 < t < \pi/2\) and \(\pi/2 < t < 3\pi/2\), these functions produce alternating signs, depicting the separate branches.
- The range \(-\pi/2 < t < \pi/2\) corresponds to one branch.- The range \(\pi/2 < t < 3\pi/2\) corresponds to the other branch.
Identifying these domains for \(t\) helps in knowing where each branch of the hyperbola lies and how the entire curve spreads over the coordinate plane, giving a comprehensive view of its geometric form.