Problem 11 - Precalculus Mathematics for Calculus

Question

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$9 x^{2}+4 y^{2}=36$$

Step-by-Step Solution

Answer

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

Step 1: Identify the standard form of the ellipse equation

First, we need to write the equation in the standard form of an ellipse, which is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) or \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\). To do this, divide everything by 36 to get the equation in the form of \(\frac{x^2}{4} + \frac{y^2}{9} = 1\).

Step 2: Determine the orientation of the ellipse

Compare the coefficients \(\frac{x^2}{4}\) and \(\frac{y^2}{9}\). Since \(b^2 = 9 > a^2 = 4\), the equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\) represents a vertical ellipse with the major axis along the y-axis.

Step 3: Identify the lengths of the major and minor axes

For the vertical ellipse, the length of the major axis is determined by \(2b\), and the length of the minor axis is \(2a\). Here, \(b = 3\) and \(a = 2\), so the major axis is \(2 \times 3 = 6\), and the minor axis is \(2 \times 2 = 4\).

Step 4: Find the vertices of the ellipse

The vertices for a vertical ellipse centered at the origin are \((0, \pm b)\). With \(b = 3\), the vertices are at \((0, 3)\) and \((0, -3)\).

Step 5: Calculate the foci of the ellipse

The foci are located at \((0, \pm c)\) for a vertical ellipse, where \(c = \sqrt{b^2 - a^2}\). Here, \(c = \sqrt{9 - 4} = \sqrt{5}\), so the foci are \((0, \sqrt{5})\) and \((0, -\sqrt{5})\).

Step 6: Determine the eccentricity of the ellipse

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

Step 7: Sketch the graph of the ellipse

Draw a coordinate plane and plot the center at \((0, 0)\), the vertices at \((0, 3)\) and \((0, -3)\), and the points corresponding to the length of the minor axis (\((2, 0)\) and \((-2, 0)\)). Sketch the ellipse shape passing through these four points.

Key Concepts

Vertices of an Ellipse

The vertices of an ellipse are the two farthest points on the ellipse along the major axis. They represent crucial points for understanding the size and shape of the ellipse.
In our case, we have the equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \), which shows a vertical ellipse because the larger denominator is under the \( y^2 \) term. This means the major axis is aligned with the y-axis.
For this vertical ellipse, the vertices are positioned at \((0, \pm b)\). Here, \(b = 3\), thus making our vertices \((0, 3)\) and \((0, -3)\). These points are directly above and below the center at the origin \((0, 0)\) and help define the elongation of the ellipse along the y-axis.

Foci of an Ellipse

The foci (plural of focus) of an ellipse are two special points located along the major axis. The sum of the distances from any point on the ellipse to both foci is constant. This property plays a key role in understanding the geometry of ellipses.
For the example equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \), the foci are situated at \((0, \pm c)\), due to the vertical orientation, where \(c\) is the focal distance computed from \(c = \sqrt{b^2 - a^2}\). Here, \(b^2 = 9\) and \(a^2 = 4\), so \(c = \sqrt{9 - 4} = \sqrt{5}\).
Therefore, the foci are located at \((0, \sqrt{5})\) and \((0, -\sqrt{5})\). The presence of these points signifies where the curvature of the ellipse is decidedly influenced.

Eccentricity of an Ellipse

The eccentricity of an ellipse is a measure that describes how much the shape of the ellipse deviates from being a circle. It determines the degree of elongation.
The eccentricity \(e\) is calculated using the formula \(e = \frac{c}{b}\), where \(c\) is the focal distance and \(b\) is the semi-major axis length.
In our problem, with \(c = \sqrt{5}\) and \(b = 3\), the eccentricity \(e\) becomes \(\frac{\sqrt{5}}{3}\).
This value indicates a mildly elongated shape along the major axis, as any ellipse with eccentricity between 0 and 1 is stretched. An eccentricity close to 0 means it is almost circular, whereas values closer to 1 indicate a more pronounced ellipse shape.

Major and Minor Axes

The major and minor axes are the longest and shortest diameters of the ellipse, respectively. These axes cross at the center of the ellipse and are perpendicular to each other.
In a vertical ellipse like in our equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \), the major axis is along the y-axis, making \(b = 3\) the number to determine its length.

• Major Axis: It is twice the length of \(b\), which is \(2b = 6\).
• Minor Axis: It is twice the length of \(a\), which is \(2a = 4\).

These axes not only define the horizontal and vertical spread of the ellipse but also help to quickly assess the eccentricity and overall balance of the shape.