Problem 31 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The standard form of the rectangular equation for the ellipse is \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\).

Step 1: Identify the given equations

The two given equations are \(x=h+a\ \cos\ \theta\) and \(y=k + b\ \sin\ \theta\). Here, \(x\) and \(y\) are the coordinates of the points on the ellipse, with \(\theta\) as the parameter. \(h\) and \(k\) are the coordinates of the center of the ellipse, while \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.

Step 2: Use trigonometric identity

We need to eliminate parameter \(\theta\) from the given equations. Using the Pythagorean trigonometric identity \(\cos^2 \theta + \sin^2 \theta = 1\), we can rearrange the given equations to be in the form of \(cos \theta\) and \(sin \theta\). By doing so, we get \(\cos \theta = (x-h)/a\) and \(\sin\ \theta = (y-k)/b\).

Step 3: Substitute and simplify

Now, substitute \(\cos \theta\) and \(\sin \theta\) from step 2 into the identity. This gives: \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\). This is the standard form of the equation of an ellipse.

Key Concepts

Ellipse

An ellipse is a captivating geometric shape that resembles an elongated circle. It is formed by all points that are the sum of the distances to two fixed points, known as the foci, is constant. The standard form of an ellipse is represented as an equation in the Cartesian plane, which could be in the form of

• \[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\]

where

• \(h\) and \(k\) are the coordinates of the center of the ellipse,
• \(a\) is the semi-major axis length, and
• \(b\) is the semi-minor axis length.

If \(a > b\), the ellipse is stretched further along the horizontal axis, whereas \(a < b\) indicates that the ellipse is stretched vertically. Understanding the basic properties of an ellipse helps in visualizing the shape and solving problems related to its equation.

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are always true for any value of the variables involved. One of the most fundamental trigonometric identities is the Pythagorean identity:

• \[\cos^2 \theta + \sin^2 \theta = 1.\]

This identity is derived from the Pythagorean theorem and is essential when dealing with relationships on the unit circle. In solving the given problem, this identity is used to eliminate the trigonometric parameter \(\theta\) from the parametric equations of the ellipse.
By transforming parametric equations into non-parametric forms using identities like these, we can better understand and manipulate the functions involved. It simplifies expressions and aids in converting parameter-dependent equations into rectangular (Cartesian) formats.

Parameter Elimination

Parameter elimination is a valuable technique for converting parametric equations into their equivalent rectangular form. In parametric equations, a parameter such as \(\theta\) is introduced, which helps to define a position on a geometric figure through functions of one variable. The goal of parameter elimination is to remove this parameter to express the relationship solely in terms of the rectangular coordinates, \(x\) and \(y\).
In the context of the ellipse problem:

• We have two parametric equations given for \(x\) and \(y\) in terms of \(\theta\),
• By using the trigonometric identity \(\cos^2 \theta + \sin^2 \theta = 1\), we can express \(\cos \theta\) and \(\sin \theta\) directly in terms of \(x\) and \(y\),
• Substituting these expressions back into the identity allows \(\theta\) to be eliminated, resulting in a compact rectangular equation format.

This process reveals the deep-seated relations between trigonometric and algebraic representations, simplifying complex geometric problems into more manageable algebraic forms.