Problem 77 - Calculus Early Transcendentals

Question

Find parametric equations (not unique) of the following ellipses (see Exercises \(75-76\) ). Graph the ellipse and find a description in terms of \(x\) and \(y.\) An ellipse centered at the origin with major axis of length 6 on the \(x\) -axis and minor axis of length 3 on the \(y\) -axis, generated counterclockwise

Step-by-Step Solution

Answer

Answer: The parametric equations for the given ellipse are \(x(t) = 3\cos(t)\) and \(y(t) = 1.5\sin(t)\). The standard equation of the ellipse is \(\frac{x^2}{9} + \frac{y^2}{2.25} = 1\).

Step 1: Identify the values of a and b

For this ellipse, the major axis length is 6, and the minor axis length is 3. The lengths of the major and minor axes are \(2a\) and \(2b\), respectively. Therefore, we can find the values of a and b as follows: \(2a = 6 \Rightarrow a = 3\) \(2b = 3 \Rightarrow b = 1.5\)

Step 2: Write the parametric equations

Now that we have the values of a and b, we can write the parametric equations for this ellipse. The parametric equations are as follows: \(x(t) = a\cos(t)\) \(y(t) = b\sin(t)\) In this case: \(x(t) = 3\cos(t)\) \(y(t) = 1.5\sin(t)\)

Step 3: Find a description in terms of x and y

Finally, let's find a description of the ellipse in terms of x and y. To do this, we can substitute the parametric equations into the standard ellipse equation like so: \(\frac{(3\cos(t))^2}{(3)^2} + \frac{(1.5\sin(t))^2}{(1.5)^2} = 1\) Simplifying gives: \(\frac{9\cos^2(t)}{9} + \frac{2.25\sin^2(t)}{2.25} = 1\) \(\cos^2(t) + \sin^2(t) = 1\) Since \(\cos^2(t) + \sin^2(t)\) is always equal to 1, the equation simplifies to: \(\frac{x^2}{9} + \frac{y^2}{2.25} = 1\)

Key Concepts

Understanding an Ellipse

An ellipse is a geometric shape that looks like a stretched circle. It has two main dimensions, or axes: the major and minor axes. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest. Understanding these axes is crucial since they define the size and shape of the ellipse.

• The ends of the major axis are known as the vertices of the ellipse.
• Both axes are perpendicular to each other and intersect at the ellipse's center.
• In this exercise, the ellipse is centered at the origin—meaning its center is at the point (0,0) on a graph.

The ellipse has a formula called the standard equation of an ellipse, which we will discuss in another section. But for now, know that the values of the major and minor axes directly affect this formula. The nature of an ellipse makes it a fascinating figure with extensive application in fields like astronomy where planets follow elliptical orbits.

Trigonometric Functions and Parametric Equations

Trigonometric functions such as cosine (\( \cos(t)\)) and sine (\( \sin(t)\)) are tools that help describe circles and ellipses. These functions are crucial when using parametric equations to express an ellipse's path. Trigonometric functions take an angle as input, often measured in radians or degrees, and output a value representing a point on a unit circle.
Parametric equations use these outputs to define an ellipse:

• \( x(t) = a \cos(t) \) - where \( a \) is half the length of the major axis.
• \( y(t) = b \sin(t) \) - where \( b \) is half the length of the minor axis.

In the given exercise, the parametric equations are:

• \( x(t) = 3\cos(t) \)
• \( y(t) = 1.5\sin(t) \)

These equations describe how a point travels around the boundary of the ellipse as you vary the angle \( t \) from \( 0 \) to \( 2\pi \) radians (a full circle). Each value of \( t \) gives us a specific point on the ellipse.

The Standard Form of an Ellipse

The standard form of an ellipse equation provides a general relationship between the ellipse's horizontal and vertical stretches. This equation reveals details about the ellipse's size and how it is oriented on a graph. The standard form equation of an ellipse centered at the origin with horizontal and vertical axes is:

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

In this equation:

• \( a \) is half the length of the major axis (horizontal stretch).
• \( b \) is half the length of the minor axis (vertical stretch).

From the solution, we know:

• \( a = 3 \)
• \( b = 1.5 \)

Substituting these values, the equation becomes:

• \( \frac{x^2}{9} + \frac{y^2}{2.25} = 1 \)

This equation states that however you choose the points \( (x, y) \)on the ellipse, when substituted into this equation, it will hold true. This characteristic shows why ellipses are part of the conic sections, and expressing them this way is crucial for analysis and geometric interpretations.