Problem 22 - Precalculus: Mathematics for Calculus

Question

An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$x^{2}+3 y^{2}=9$$

Step-by-Step Solution

Answer

Vertices: \((3, 0), (-3, 0)\); Foci: \((\sqrt{6}, 0), (-\sqrt{6}, 0)\); Eccentricity: \(\frac{\sqrt{6}}{3}\). Major axis length: 6, Minor axis: \(2\sqrt{3}\).

Step 1: Transform to Standard Form

The given equation is \(x^2 + 3y^2 = 9\). First, divide every term by 9 to get \(\frac{x^2}{9} + \frac{y^2}{3} = 1\). This is the standard form of an ellipse with \(a^2 = 9\) and \(b^2 = 3\).

Step 2: Identify the Major and Minor Axes

In the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), if \(a^2 > b^2\), then the ellipse is oriented horizontally. Here, \(a^2 = 9\) and \(b^2 = 3\), so \(a = 3\) and \(b = \sqrt{3}\). Therefore, the major axis is along the x-axis.

Step 3: Calculate the Vertices

The vertices of an ellipse aligned along the x-axis are at \(\pm a\), which gives \(\pm 3\). So the vertices are \((3, 0)\) and \((-3, 0)\).

Step 4: Find the Foci

The foci are found using \(c^2 = a^2 - b^2\). Here, \(c^2 = 9 - 3 = 6\), so \(c = \sqrt{6}\). The foci are \((\pm \sqrt{6}, 0)\).

Step 5: Calculate Eccentricity

The eccentricity \(e\) is given by \(e = \frac{c}{a}\). Therefore, \(e = \frac{\sqrt{6}}{3}\).

Step 6: Lengths of the Axes

The length of the major axis is \(2a = 6\), and the length of the minor axis is \(2b = 2\sqrt{3}\).

Step 7: Sketch the Ellipse

To sketch the ellipse, plot the vertices \((3, 0)\) and \((-3, 0)\), draw the major axis of length 6 along the x-axis, and the minor axis of length \(2\sqrt{3}\) along the y-axis. Plot the foci at \((\sqrt{6}, 0)\) and \((-\sqrt{6}, 0)\) for guidance.

Key Concepts

Vertices of an ellipse

The vertices of an ellipse are crucial points that define its width and shape. For an ellipse defined in standard form as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the vertices help us understand the extent of the ellipse along its major axis. When the ellipse is oriented horizontally, the vertices are located at coordinates \( (\pm a, 0) \) if the major axis is along the x-axis.
In our given equation, where the standard form is \( \frac{x^2}{9} + \frac{y^2}{3} = 1 \), we have that \( a^2 = 9 \), which means \( a = 3 \). Therefore, there are two vertices located at coordinates \( (3, 0) \) and \( (-3, 0) \). These points mark the maximum horizontal distance across the ellipse.

Foci of an ellipse

The foci of an ellipse are two fixed points located along the major axis, which play a key role in its geometric structure. You can find the foci using the relationship \( c^2 = a^2 - b^2 \). For an ellipse in the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the foci are at \( (\pm c, 0) \) if the major axis is horizontal.
In our example, with \( a^2 = 9 \) and \( b^2 = 3 \):

• Calculate \( c^2 = 9 - 3 = 6 \).
• Find \( c = \sqrt{6} \).

The foci are located at coordinates \( (\sqrt{6}, 0) \) and \( (-\sqrt{6}, 0) \), inside the ellipse near its center.

Eccentricity of an ellipse

Eccentricity is a numerical measure that describes how "stretched" the shape of an ellipse is. It's denoted by the symbol \( e \) and calculated using the formula \( e = \frac{c}{a} \). It provides insight into the ellipse's form:

• An eccentricity of \( e = 0 \) indicates a perfect circle.
• An eccentricity approaching \( e = 1 \) suggests a very elongated shape.

In this problem, we have:

• \( a = 3 \)
• \( c = \sqrt{6} \)

Thus, the eccentricity is \( e = \frac{\sqrt{6}}{3} \). This value tells us that the ellipse is elongated, but not incredibly so, which is typical for most ellipses.

Major and minor axes of an ellipse

The major and minor axes of an ellipse define its dimensions and orientation. The major axis is the longest diameter, running through the foci and vertices, while the minor axis is perpendicular to it and is shorter.
For the ellipse equation \( \frac{x^2}{9} + \frac{y^2}{3} = 1 \):

• Major axis length: \( 2a = 6 \)
• Minor axis length: \( 2b = 2\sqrt{3} \)

The major axis lies along the x-axis, as \( a^2 > b^2 \), emphasizing the extended width of the ellipse. The minor axis, being vertical, highlights its height.

Standard form of an ellipse

The standard form of an ellipse is a simplified expression that allows easy identification and analysis of its properties. It's generally given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This form makes it straightforward to discern important characteristics like:

• Orientation of the ellipse: Horizontal if \( a > b \), vertical if \( b > a \).
• The values of \( a^2 \) and \( b^2 \), which are key to calculating vertices and axes lengths.

In the exercise, starting from \( x^2 + 3y^2 = 9 \), we divided every term by 9 to get it into standard form. This process results in \( \frac{x^2}{9} + \frac{y^2}{3} = 1 \), revealing a horizontally oriented ellipse. This transformation is crucial as it sets the groundwork for subsequent analyses.