Problem 38 - Precalculus: Functions and Graphs

Question

Show that $$\begin{array}{c}x=a \sec t+h, \quad y=b \tan t+k -\pi / 2

Step-by-Step Solution

Answer

The hyperbola has center \((h, k)\), transverse axis length \(2a\), conjugate axis length \(2b\). Right branch: \(-\pi/2 < t < \pi/2\); left branch: \(\pi/2 < t < 3\pi/2\).

Step 1: Identify Parametric Equations

The given parametric equations are \( x = a \sec(t) + h \) and \( y = b \tan(t) + k \). These are the forms that we need to analyze to derive the equation of the curve.

Step 2: Rearrange and Eliminate Parameter

From the parametric equations, express \( \sec(t) \) and \( \tan(t) \) in terms of \( x \) and \( y \):\[\sec(t) = \frac{x - h}{a}, \quad \tan(t) = \frac{y - k}{b}.\] Use the identity \( \sec^2(t) - \tan^2(t) = 1 \) to eliminate \( t \).

Step 3: Apply Hyperbola Identity

Substitute the expressions for \( \sec(t) \) and \( \tan(t) \) into the identity:\[\left( \frac{x-h}{a} \right)^2 - \left( \frac{y-k}{b} \right)^2 = 1.\] This equation is of the form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \), which is the standard form of a hyperbola.

Step 4: Identify Hyperbola Characteristics

The standard form equation identified in Step 3 confirms that the hyperbola has a center at \((h,k)\). The transverse axis is aligned with \(x\) and has length \(2a\). The conjugate axis is aligned with \(y\) and has length \(2b\).

Step 5: Determine Range for Each Branch

Given \(-\pi/2 < t < 3\pi/2\) and \(t eq \pi/2\), these bounds for \(t\) need to be split to identify the branches caused by the restrictions implied by \( \sec(t) \) and \( \tan(t) \). The branch where \( \sec(t) \) is positive (right branch) corresponds to \(-\pi/2 < t < \pi/2\). The branch for the negative part (left branch) corresponds to \(\pi/2 < t < 3\pi/2\).

Key Concepts

Hyperbola

A hyperbola is one of the intriguing shapes in conic sections, known for its two distinct branches. Imagine slicing through a double cone; when you cut at a steep angle parallel to the axis, you get a hyperbola. It's a bit like slicing an apple in such a way that you end up with two curved pieces. Each branch of the hyperbola curves away, creating an open shape.

The standard mathematical form of a hyperbola is expressed as \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\].
Here, \(h\) and \(k\) are coordinates of the center, while \(a\) and \(b\) are distances that define the size of the hyperbola's axes. The transverse axis, the longer path across its center, has a length of \(2a\). Meanwhile, the conjugate axis is perpendicular and measures \(2b\).

This standard equation helps in visualizing and analyzing hyperbolas in algebraic terms, allowing us to understand their orientation and dimensions more deeply.

Secant and Tangent Functions

The secant and tangent functions play a vital role in describing angular aspects of trigonometry just as much as they reveal about circular and periodic functions. The secant function, denoted as \(\sec(t)\), is essentially the reciprocal of the cosine function: \(\sec(t) = \frac{1}{\cos(t)}\).

Meanwhile, the tangent function, \(\tan(t) = \frac{\sin(t)}{\cos(t)}\), accounts for the ratio of sine to cosine. When analyzing curves like hyperbolas, these functions help bridge gaps between angles and coordinates, transforming polar perspectives into rectangular coordinates. In this exercise, they are harnessed in the form \(x = a \sec(t) + h\) and \(y = b \tan(t) + k\), linking trigonometric identities with parametric definitions. This link transforms the world of angles into geometric shapes, allowing us to explore these fascinating curves more deeply.

Conic Sections

Conic sections represent a family of curves that impeccably rationalize the geometric results of slicing a cone. These curves include the circle, ellipse, parabola, and hyperbola, all having their distinct features and equations. The hyperbola, as shown in this exercise, forms one category of conic sections drawn when a plane cuts through both halves of a double cone.

Conic sections intricately map out different scenarios based on angles and intersections of a plane with a cone. For example:

• The circle arises when the cut is perpendicular to the cone's axis.
• An ellipse forms when the cut angles upwards but doesn’t cross the base.
• The parabola appears when the cut is parallel to the cone’s side.
• The hyperbola is the result when the cut is steep enough to pass through both cones.

These sections embody the immersive connection between geometry and algebra, offering intuition and insights that transcend pure mathematical formulae.

Elimination of Parameters

Elimination of parameters is a method used to convert parametric equations into a single equation, allowing us to directly compare variables without intermediary steps. It involves removing the parameter (often a variable like \(t\) or \(\theta\)) to establish direct algebraic relationships.

In this exercise, the parameter to eliminate is \(t\). The given equations are \(x = a \sec(t) + h\) and \(y = b \tan(t) + k\). By utilizing the trigonometric identity \(\sec^2(t) - \tan^2(t) = 1\), we eliminate \(t\), thus simplifying the relation between \(x\) and \(y\).
The result, \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), directly attributes curves to their equations, like solving a puzzle by clearing away unnecessary pieces. Elimination of parameters is a powerful technique in calculus and algebra, providing clarity by reducing complexity in dealing with curves.