Problem 41 - Precalculus

Question

Find the center, vertices, foci, and eccentricity of the ellipse. Then sketch the ellipse. $$9 x^{2}+4 y^{2}+36 x-24 y+36=0$$

Step-by-Step Solution

Answer

The center of the ellipse is (-2, 3). The vertices are at (-2, 0) and (-2, 6). The foci are located at (-2, 3 ± √5). Eccentricity of the ellipse is √5/3.

Step 1: Write the Equation in Standard Form

The standard form of an ellipse is \((x - h)^2 / a^2 + (y - k)^2 / b^2 = 1\), where (h, k) is the center of the ellipse, a is the distance from the center to the outer vertices (along the x-axis), and b is the distance from the center to the inner vertices (along the y-axis). Notice that the coefficients of \(x^2\) and \(y^2\) in the given equation are not equal. Divide through by the constant term so the coefficients of the squared terms become 1. In this case, divide the whole equation by 36 to get the equation on standard form. The resulting equation is \((x + 2)^2 / 4 + (y - 3)^2 / 9 = 1\).

Step 2: Find the Center, Vertices and Foci of the Ellipse

The center of the ellipse is at (-2, 3). The major axis has length 2a = 2*sqrt(9) = 6, so the vertices are (-2, 3±3), which simplifies to (-2, 0) and (-2, 6). The minor axis has length 2b = 2*sqrt(4) = 4, but there is no need to find the 'y' coordinates of these vertices since they are not asked for. Using the formula for the distance from the center to the foci \(c=\sqrt{a^2 - b^2}\), we have \(c=\sqrt{9-4}= \sqrt{5}\), so the coordinates of the foci are (-2, 3 ± √5).

Step 3: Finding the Eccentricity

The eccentricity of the ellipse is the ratio of the distance from the center to a focus over the distance from the center to a vertex. Therefore, the eccentricity e = c/a = √5/3

Step 4: Sketch the Ellipse

Plot the center, vertices, and foci on a set of coordinate axes, using the 'y' value equals 3 as the center line for the 'y' axis. Draw a smooth curve to form the ellipse. The major axis is vertical. Make sure your foci are inside the ellipse and your vertices are on the ellipse.

Key Concepts

Conic Sections

Conic sections are the curves obtained when a plane cuts through a double-napped cone at various angles and positions. These shapes include circles, ellipses, parabolas, and hyperbolas. Each type of conic section has unique properties defined by the intersection's angle.
For instance, if the plane cuts parallel to the cone's base, you get a circle. Slicing the cone at an angle, but still parallel to one of its sides, results in an ellipse. A parabola occurs when the plane is parallel to the slope of the cone, while a hyperbola appears when the plane cuts through both naps of the cone at an angle.
Conic sections are essential in the fields of astronomy, physics, engineering, and mathematics, as they describe the paths of celestial bodies and the shapes of satellite dishes and reflectors. Teaching these concepts encourages a deeper understanding of how geometric shapes are foundational in the world around us.

Standard Form of an Ellipse

An ellipse is a set of all points in a plane where the sum of the distances to two fixed points, called foci, is constant. When describing an ellipse with an equation, the standard form is \[\begin{equation}(x-h)^2/a^2 + (y-k)^2/b^2 = 1\text{.}\end{equation}\]

Identifying Ellipse Components

• Center (h, k): The middle point of the ellipse, from which distances to other parts of the ellipse are often measured.
• Major axis and semimajor axis (a): The longest diameter and its half, respectively. The semimajor axis covers the distance from the center to the furthest edge on the ellipse.
• Minor axis and semiminor axis (b): The shortest diameter and its half, respectively. The semiminor axis measures from the center to the closest edge.

By completing the square on the given equation and rearranging, we can find the center and lengths of the major and minor axes, thus fully describing the ellipse's size and position.

Eccentricity of an Ellipse

The eccentricity of an ellipse is a numerical value that describes the shape of the ellipse—specifically, how 'stretched out' it is. It's defined as the ratio of the distance from the center to a focus (\[\begin{equation}c\text{)}\end{equation}\] over the distance from the center to a vertex on the major axis (\[\begin{equation}a\text{)}:\end{equation}\] \[\begin{equation}e = c/a\text{.}\end{equation}\] The eccentricity has a range of values from 0 to 1, where 0 represents a perfect circle, and values closer to 1 indicate an increasingly elongated ellipse.
For the given ellipse in the exercise, the eccentricity is calculated using the formula for \[\begin{equation}c=\end{equation}\] \[\begin{equation}\sqrt{a^2 - b^2},\end{equation}\] which gives us a numerical measure of its shape. With an eccentricity less than 1, the ellipse shows a balance between circular and elongated forms, which is visually represented in the graph of the equation. Understanding the eccentricity helps in the analysis of elliptical orbits in celestial mechanics, such as planets orbiting the sun.