Problem 15 - 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. $$9 x^{2}+4 y^{2}=36$$

Step-by-Step Solution

Answer

Vertices: (0, 3), (0, -3); Foci: (0, ±√5); Eccentricity: √5/3; Major Axis: 6; Minor Axis: 4.

Step 1: Convert to Standard Form

The given equation is \(9x^2 + 4y^2 = 36\). First, divide the entire equation by 36 to convert it to the standard form of an ellipse.\[\frac{x^2}{4} + \frac{y^2}{9} = 1\] This represents an ellipse with the major axis along the y-axis because the larger denominator is under \(y^2\).

Step 2: Identify Vertices

For the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a^2 > b^2\), the vertices of the ellipse are at \((0, \pm b)\). Here, \(a^2 = 4\) and \(b^2 = 9\), so \(b = \pm 3\).The vertices are at \((0, 3)\) and \((0, -3)\).

Step 3: Find the Foci

The distance to the foci from the center is given by \(c = \sqrt{b^2 - a^2}\). Substituting the values, \(c = \sqrt{9 - 4} = \sqrt{5}\).The foci are at \((0, \pm \sqrt{5})\).

Step 4: Calculate Eccentricity

The eccentricity \(e\) of an ellipse is given by \(e = \frac{c}{b}\). Substituting the values, \(e = \frac{\sqrt{5}}{3}\).

Step 5: Determine Lengths of Axes

The length of the major axis is \(2b = 2 \times 3 = 6\), and the length of the minor axis is \(2a = 2 \times 2 = 4\).

Step 6: Sketch the Ellipse

Draw a coordinate plane. Plot the center at the origin (0, 0). From the center, mark the vertices at \((0, 3)\) and \((0, -3)\) on the y-axis. Mark co-vertices at \((\pm 2, 0)\) on the x-axis to guide your sketch. Draw the ellipse enclosing these points, with the longer axis (length 6) vertical.

Key Concepts

Vertices of an Ellipse

Vertices are key points located on the major axis of an ellipse. These points are where the ellipse is the widest, and they determine the span of the ellipse from top to bottom or side to side, depending on the orientation. For the ellipse expressed in its standard form \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] the vertices are found by locating \(b\), the semi-major length if \(b^2 > a^2\). The vertices will be at \((0, \pm b)\) or \((\pm b, 0)\) depending on whether the major axis is vertical or horizontal. In our example,

• The equation is \(\frac{x^2}{4} + \frac{y^2}{9} = 1\).
• The value \(b = 3\) because \(b^2 = 9\), and the minor axis has \(a = 2\) because \(a^2 = 4\).
• The vertices are at \((0, 3)\) and \((0, -3)\), as this ellipse is taller than it is wide.

Foci of an Ellipse

The foci of an ellipse are two fixed points located along the major axis. These points are crucial because the sum of the distances from any point on the ellipse to the foci is constant. To find the foci, we use the formula: \[ c = \sqrt{b^2 - a^2} \]Where \(c\) is the distance from the center of the ellipse to each focus. In our example:

• Given \(a^2 = 4\) and \(b^2 = 9\), substituting these into the formula gives \(c = \sqrt{9 - 4} = \sqrt{5}\).
• The foci are thus located at \((0, \pm \sqrt{5})\), reflecting a vertical alignment along the major axis.

These points lie inside the ellipse, and help define its unique shape.

Eccentricity of an Ellipse

The eccentricity of an ellipse is a measure of how "stretched out" the ellipse is compared to a circle. It is defined as:\[ e = \frac{c}{b} \]Where \(c\) is the distance from the center to a focus, and \(b\) is the distance from the center to a vertex on the major axis. In our case:

• We have already determined \(c = \sqrt{5}\) and \(b = 3\).
• Substituting these values gives us an eccentricity, \(e = \frac{\sqrt{5}}{3}\).

The eccentricity is always a number between 0 and 1 for ellipses. A smaller eccentricity means the ellipse is closer to being a circle.

Major and Minor Axes

The axes of an ellipse are two mutually perpendicular lines that define the longest and shortest dimensions of the ellipse. These are:

• The **Major Axis**: This is the longer axis, reaching from vertex to vertex. It goes through the center and the foci. For our ellipse, this axis is along the y-axis.
• The **Minor Axis**: This shorter axis is perpendicular to the major axis and stretches across the co-vertices. For our ellipse, it's on the x-axis.

The lengths of these axes are twice the lengths of the semi-major and semi-minor axes:

• Major Axis length = \(2b = 6\) (as \(b = 3\)).
• Minor Axis length = \(2a = 4\) (as \(a = 2\)).

This shape results in a taller ellipse, with the major axis stretching vertically.

Standard Form of an Ellipse

To analyze and graph an ellipse easily, it should be expressed in its standard form:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]The term with the larger denominator \(b^2\) indicates the direction of the major axis. The provided equation\[ 9x^2 + 4y^2 = 36 \] was converted by dividing each term by 36.

• The result is \(\frac{x^2}{4} + \frac{y^2}{9} = 1 \), placing the major axis along the vertical y-axis because \(b^2 > a^2\).
• This form allows us to directly read the values for the semi-axis lengths: \(a = 2\) and \(b = 3\).

This setup simplifies finding other ellipse characteristics like vertices, lengths of axes, eccentricity, and foci.