Question

Sketch the 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. $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$

Short answer

Answer: The vertices are at (3, 0), (-3, 0), (0, 2), and (0, -2). The foci are approximately at (2.24, 0) and (-2.24, 0). The lengths of the major and minor axes are 6 and 4 units, respectively.

Step-by-step solution

Identify Major and Minor Axes

The given ellipse is in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with \(a^2 = 9\) and \(b^2 = 4\). The major axis (\(a\)) is larger than the minor axis (\(b\)), so we have a horizontal ellipse. By taking the square roots of these values, we find that \(a = 3\) and \(b = 2\). The lengths of the major and minor axes are twice these values, which are 6 and 4 units, respectively.

Find Vertices

To find the vertices, we go a distance of \(a = 3\) units to the left and right of the center (0, 0) along the major axis, and a distance of \(b = 2\) units up and down of the center (0, 0) along the minor axis. The vertices are at points: $$(\pm a, 0) \Rightarrow (\pm 3, 0) \Rightarrow (3, 0)\text{ and }(-3, 0)$$ $$(0, \pm b) \Rightarrow (0, \pm 2) \Rightarrow (0, 2)\text{ and }(0, -2)$$

Find Foci

To find the foci of the ellipse, we use the relationship \(c^2 = a^2 - b^2\), where \(c\) is the distance from the center of the ellipse to each focus. Here, \(a^2 = 9\) and \(b^2 = 4\), so \(c^2 = 9 - 4 = 5\). Taking the square root, we find \(c = \sqrt{5}\). The foci will also lie along the major axis so their coordinates are given by \((\pm c, 0) \Rightarrow (\pm \sqrt{5}, 0) \Rightarrow (\sqrt{5}, 0)\text{ and }(-\sqrt{5}, 0)\).

Plot the Graph

Now that we have the vertices, foci, and lengths of the major and minor axes, we can sketch the graph of the ellipse. Place the center at the origin (0, 0). Plot the vertices at (3, 0), (-3, 0), (0, 2), and (0, -2), and the foci at approximately (2.24, 0) and (-2.24, 0). Draw the horizontal major axis of length 6 units and the vertical minor axis of length 4 units. Sketch the ellipse with a smooth curve that goes through the vertices and hugs the foci. Finally, check your work with a graphing utility. The graph of the ellipse is now complete, along with the coordinates of the vertices, foci, and the lengths of the major and minor axes.

Key concepts

Ellipse Equation

An ellipse is a curved shape that can be drawn on the coordinate plane using its equation. The general equation for an ellipse centered at the origin \( \left(0,0\right) \) is:

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

Here, \( a^2 \) and \( b^2 \) determine the size and orientation of the ellipse. If \( a > b \), the ellipse is wider horizontally and the term \( a^2 \) is associated with the major axis. Conversely, if \( b > a \), the ellipse is taller vertically and \( b^2 \) is associated with the major axis.
In our example, \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), where \( a^2 = 9 \) and \( b^2 = 4 \). Here, \( a^2 \) is larger, signifying a horizontal major axis.
By understanding and analyzing this base equation, you can derive more specific characteristics of the ellipse such as its vertices and foci, which further define its dimensions and position.

Vertices of Ellipse

The vertices of an ellipse are key points where the ellipse intersects its major and minor axes. Finding these points lets us define the ellipse's shape and size.
For an ellipse centered at the origin \((0,0)\), the vertices along the major axis are located at \(( \pm a, 0)\) if it's horizontal. Along the minor axis, they are at \((0, \pm b)\). By substituting the values of \( a \) and \( b \), we determine their actual positions.
In this exercise:

• Major axis vertices: \((\pm 3, 0)\) gives us \((3, 0)\) and \((-3, 0)\).
• Minor axis vertices: \((0, \pm 2)\) gives us \((0, 2)\) and \((0, -2)\).

These coordinates fully define the outer bounds of the ellipse within its axes of symmetry.

Foci of Ellipse

The foci of an ellipse provide insights into how stretched or elongated the ellipse is. They lie along the major axis, equidistant from the center.
To find the distance of each focus from the center, we use the formula:

• \( c^2 = a^2 - b^2 \)

For the given problem:

• \( a^2 = 9 \)
• \( b^2 = 4 \)

Thus,\( c^2 = 9 - 4 = 5 \), and \( c = \sqrt{5} \approx 2.24 \).
The foci are positioned at \((\pm \sqrt{5}, 0)\) which converts to approximately \((2.24, 0)\) and \((-2.24, 0)\). These points lie within the boundaries outlined by the major axis, giving the ellipse its unique shape.

Major and Minor Axes

The major and minor axes are essential in defining the overall dimensions of the ellipse. These axes are perpendicular to each other, intersecting at the ellipse's center, either at the origin \((0, 0)\) or another specified point for ellipses not centered at the origin.
For a horizontal ellipse, the major axis is longer and runs along the x-axis, while the minor axis is shorter and runs along the y-axis.
In our example:

• **Major axis:** The length is twice \( a \). Given \( a = 3 \), its length is \( 2a = 6 \) units.
• **Minor axis:** The length is twice \( b \). Given \( b = 2 \), its length is \( 2b = 4 \) units.

These axes not only determine the ellipse's dimensions but also influence its overall shape.
Thus, when graphing an ellipse, recognizing these measurements helps ensure accurate positioning and scale on the coordinate plane.