Problem 31 - Calculus Early Transcendentals

Question

Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$\frac{x^{2}}{5}+\frac{y^{2}}{7}=1$$

Step-by-Step Solution

Answer

Answer: The coordinates of the vertices on the major axis are $(-\sqrt{7}, 0)$ and $(\sqrt{7}, 0)$, and the coordinates of the vertices on the minor axis are $(0, -\sqrt{5})$ and $(0, \sqrt{5})$. The coordinates of the foci are $(-\sqrt{2}, 0)$ and $(\sqrt{2}, 0)$.

Step 1: Determine the center

The given equation is $$\frac{x^2}{5} + \frac{y^2}{7} = 1$$ Here, we can see that h=0 and k=0 as the equation has the form \(\frac{x^2}{5}+\frac{y^2}{7}=1\) (no addition or subtraction in the numerator). So, the center of the ellipse is (0,0).

Step 2: Identify the major and minor axes

In our equation, the coefficients are \(\frac{1}{5}\) and \(\frac{1}{7}\). Since \(a^2>b^2\), we will have a horizontal major axis. First, let's find the lengths of the major and minor axes: \(a^2 = 7\), and \(b^2 = 5\). Hence, \(a= \sqrt{7}\) and \(b= \sqrt{5}\).

Step 3: Calculate the vertices

To find the vertices, we use the values of a and b as distances along the major and minor axes, respectively. Since the center is at (0,0) and the major axis is horizontal, the vertices are as follows: Vertices on the major axis: \((-\sqrt{7}, 0)\) and \((\sqrt{7}, 0)\) Vertices on the minor axis: \((0, -\sqrt{5})\) and \((0, \sqrt{5})\)

Step 4: Calculate the foci

To find the foci, we use the relation \(c=\sqrt{a^2 - b^2}\), where c is the distance of the foci from the center along the major axis. \(c= \sqrt{7 - 5}\) \(c= \sqrt{2}\). The foci will be located at distances of \(±\sqrt{2}\) away from the center along the major axis. Therefore, the coordinates of the foci are: Foci: \((-\sqrt{2}, 0)\) and \((\sqrt{2}, 0)\).

Step 5: Sketch the graph

Now that we have the center, vertices, and foci, we can sketch the graph of the ellipse. Plot the center at (0,0), the vertices at \((-\sqrt{7}, 0)\), \((\sqrt{7}, 0)\), \((0, -\sqrt{5})\), and \((0, \sqrt{5})\), and the foci at \((-\sqrt{2}, 0)\) and \((\sqrt{2}, 0)\). Then, draw an ellipse by connecting the vertices. Label each point on the graph accordingly. Use a graphing calculator or an online graphing utility to verify the sketch.

Key Concepts

Graphing Ellipses

An ellipse is a smooth, closed curve that resembles an elongated circle, much like the shape of an oval. When graphing an ellipse, the equation provides essential details about its shape and orientation.

In the standard form of an ellipse, the equation is either \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\] or \[\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\].
Here,

• \(h\) and \(k\) represent the center of the ellipse.
• \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.
• If \(a > b\), the ellipse is stretched along the x-axis, and if \(b > a\), it stretches along the y-axis.

For the equation \(\frac{x^2}{5} + \frac{y^2}{7} = 1\), we observe that it has a vertical major axis since \(b^2 \gt a^2\).
This makes the ellipse taller rather than wider. The equation tells us how far along each axis the ellipse will stretch, based on the values beneath \(x^2\) and \(y^2\).
To graph, begin by marking the center point, then determine how far the ellipse extends from the center by calculating values of \(a\) and \(b\), and plot these as guides to sketch a perfect ellipse.

Vertices and Foci

Vertices and foci are key elements that define an ellipse. They help in understanding its unique shape.

The **vertices** of an ellipse are the points where the ellipse is widest or tallest along its major axis. Using our example, the vertices for the vertical major axis are found by calculating \(b = \sqrt{7}\), making the vertices at

• \((0, \sqrt{7})\)
• \((0, -\sqrt{7})\)

For the minor axis, the vertices are located at

• \((\sqrt{5}, 0)\)
• \((-\sqrt{5}, 0)\)

**Foci** are two singular points inside the ellipse. They lie along the major axis at equal distances from the center. These points have a unique property: for any point on the ellipse, the sum of distances to the two foci is constant.
The foci of our ellipse are determined by the formula \(c = \sqrt{a^2 - b^2}\). For the given ellipse:

• \(c = \sqrt{2}\)
• The foci are at \((\sqrt{2}, 0)\) and \((-\sqrt{2}, 0)\)

In essence, the vertices mark the extent of the ellipse while the foci guide its unique shape.

Major and Minor Axes

Every ellipse has two axes: the major and minor axes, which help describe its dimension and orientation.

The **major axis** is the longest diameter that passes through the center of the ellipse, covering the endpoints (or vertices) on the ellipse. In our equation, the vertical axis is longer, making this the major axis. The length of the major axis is

• \(2b = 2 \cdot \sqrt{7}\)

The **minor axis** is the shortest diameter, which also passes through the center but is perpendicular to the major axis. The length of the minor axis is

• \(2a = 2 \cdot \sqrt{5}\)

Understanding these axes is crucial for sketching as they dictate the overall length and width of the ellipse. They also help determine how the ellipse is oriented on the coordinate plane.
This fundamental component is what makes an ellipse differ from a simple circle, which only has one radius. The positioning of these axes enables us to plot the ellipse accurately, revealing its symmetry and form.