Question

Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. An ellipse with vertices (±9,0) and eccentricity \(\frac{1}{3}\)

Short answer

Answer: The equation of the ellipse is $\frac{x^2}{81} + \frac{y^2}{72} = 1$. The foci are at (±3,0), and the directrices are given by the equations $x=±27$.

Step-by-step solution

Determine the semi-major axis

As the vertices are given at \((±9,0)\), we know that the ellipse is horizontal, and the semi-major axis (a) is 9, as it extends 9 units to the right and left of the center.

Determine the eccentricity and semi-minor axis

We are given that the eccentricity is \(\frac{1}{3}\). The formula relating the semi-major axis (a), semi-minor axis (b), and eccentricity (e) is: \(b = a\sqrt{1-e^2}\) Substituting the values, we have \(b = 9\sqrt{1-\frac{1}{3}^2}\). After calculating, we get \(b = 9\sqrt{\frac{8}{9}} = 3\sqrt{8}\)

Write the equation of the ellipse

Now that we have the semi-major axis (a), semi-minor axis (b), and center (0,0), we can write the equation of the ellipse: \(\frac{x^2}{9^2} + \frac{y^2}{(3\sqrt{8})^2} = 1\), which simplifies to \(\frac{x^2}{81} + \frac{y^2}{72} = 1\)

Find the foci

Knowing the eccentricity (e) and semi-major axis (a), we can find the distance from the center to the foci using the formula \(c = ae\). Substituting values, we get \(c = \frac{1}{3} \cdot 9 = 3\). Since the ellipse is horizontal, the foci will lie to the right and left of the center along the x-axis at points \((±3,0)\)

Find the directrices

We use the formula \(d = \frac{a}{e}\) to find the distance from the center to the directrices. Substituting values, we get \(d = \frac{9}{\frac{1}{3}} = 27\) Since the ellipse is horizontal, the directrices will be vertical lines with the equations \(x = ±27\)

Sketch the graph

Sketch the graph of the ellipse, labeling the center, vertices, foci, and directrices. The ellipse has a horizontal axis with vertices at \((±9,0)\), foci at \((±3,0)\), directrices at \(x = ±27\), and a center at \((0,0)\). Add asymptotes at \(y=\pm \frac{3\sqrt{8}}{9}x\) To check your work, use a graphing utility to plot the ellipse and verify that the sketch matches the equation.

Key concepts

Semi-Major Axis

The semi-major axis of an ellipse is one of its most fundamental characteristics. It represents half of the longest diameter of the ellipse. Think of it as the 'radii' for circles, but for ellipses, since ellipses are essentially stretched circles. In simple terms, if you think about an ellipse like an oval track, the semi-major axis would be the longer straight path that runners would take per lap.

For a given ellipse equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \(a > b\), \(a\) is the length of the semi-major axis. It's crucial for determining the shape of the ellipse; the greater the value of \(a\), the more elongated the ellipse. In our exercise, the distance from the center to the vertices along the x-axis gave us a semi-major axis of 9 units.

Eccentricity

Eccentricity is a measure that describes how much an ellipse deviates from being a circle. A circle can be seen as an ellipse with eccentricity of 0, meaning it's perfectly even all around. An eccentricity close to 1 indicates a highly elongated shape, whereas an eccentricity closer to 0 means the ellipse is more circular.

In the exercise, the given eccentricity was \(\frac{1}{3}\), which suggests a relatively round shape since it's closer to 0 than to 1. To find the semi-minor axis, we used the eccentricity in the formula \(b = a\sqrt{1-e^2}\), where \(b\) is the semi-minor axis. Understanding eccentricity is vital because it also helps us determine the locations of the foci and the directrices of the ellipse.

Directrices

The directrices of an ellipse are less commonly discussed, but they are just as important in understanding the geometry of the ellipse. These are lines that are parallel to the minor axis and are located at a distance from the center. Directrices provide a reference for constructing the ellipse and work together with foci to give an alternative definition of the ellipse involving distances.

In mathematical terms, the directrices are found at a distance \(d\) from the center, and that distance is calculated using the formula \(d = \frac{a}{e}\), where \(a\) is the semi-major axis and \(e\) is the eccentricity. In the exercise, we found the directrices to be at \(x = \pm27\), meaning they are vertical lines since our ellipse is horizontally aligned. They are a fundamental piece in the complete description of the shape and properties of the ellipse.

Conic Sections

Conic sections are the various shapes that can be obtained by slicing a cone with a plane. These shapes include circles, ellipses, parabolas, and hyperbolas, depending on the angle of the cut. An ellipse is one type of conic section, formed when the slicing plane cuts through the cone at an angle that is oblique to the base of the cone and does not intersect the apex.

This is important to understand because the properties of the ellipse—like its vertices, foci, and directrices—can all be derived from the perspective of conic sections. Each conic section has distinctive features and formulas associated with it, and ellipses in particular are known for their smooth, oval-like shape. When we refer to the ellipse equation used in the exercise, we are applying this larger concept of conic sections to a specific instance, involving specific values of a semi-major axis and an eccentricity.