Problem 52 - A Graphical Approach to College Algebra

Question

Write an equation for each conic. Each parabola has vertex at the origin, and each ellipse or hyperbola is centered at the origin. $$\text { Focus }(0,-2) ; e=\frac{2}{3}$$

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{5} + \frac{y^2}{9} = 1\).

Step 1: Identify conic type

The given conic has a focus at \((0, -2)\) and an eccentricity \(e = \frac{2}{3}\). Since \(e < 1\), this indicates that the conic is an ellipse.

Step 2: Determine the direction of the ellipse

The focus at \((0, -2)\) suggests that the major axis of the ellipse is vertical, as the ordinate (y-coordinate) changes from the center (origin) to the focus.

Step 3: Use eccentricity to find \\(a\\) and \\(b\\)

The formula for eccentricity of an ellipse is \(e = \frac{c}{a}\), where \(c\) is the distance from the center to the focus, and \(a\) is the semi-major axis. Given \(c = 2\) and \(e = \frac{2}{3}\), we find that \(a = \frac{c}{e} = \frac{2}{\frac{2}{3}} = 3\).

Step 4: Calculate \\(b\\) using the ellipse relationship

For an ellipse, the relationship between \(a\), \(b\), and \(c\) is \(b^2 = a^2 - c^2\). So we have \(b^2 = 3^2 - 2^2 = 9 - 4 = 5\), giving \(b = \sqrt{5}\).

Step 5: Write the equation of the ellipse

An ellipse with a vertical major axis and centered at the origin has the equation \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\). Substituting the values of \(a\) and \(b\), the equation becomes \(\frac{x^2}{5} + \frac{y^2}{9} = 1\).

Key Concepts

Ellipse

An ellipse is a type of conic section that appears like a stretched circle. It has two focal points, or foci, and any point on the ellipse is such that the sum of the distances to these foci is constant. Ellipses are important in geometry, astronomy, and optics. You might hear them being associated with orbits of planets and satellites.

Key properties of an ellipse include:

• Major Axis: This is the longest diameter of the ellipse, running through its center and both its foci.
• Minor Axis: Perpendicular to the major axis, it's the shortest diameter.
• Center: The midpoint of both the major and minor axes.
• Vertices: Points where the ellipse intersects the major axis.

By understanding these properties, you can better grasp how an ellipse differs from other conic sections like circles and hyperbolas.

Eccentricity

Eccentricity is a measure of how much a conic section deviates from being a circle. For ellipses, this value is always between 0 and 1. A lower eccentricity indicates a shape closer to a circle, while a higher eccentricity suggests a more elongated ellipse.

This value is a ratio given by the formula:

• \( e = \frac{c}{a} \)

where:

• e: Eccentricity
• c: Distance from the center to a focus
• a: Length of the semi-major axis

For example, if the eccentricity is \( \frac{2}{3} \), it means that the ellipse is not overly elongated but still has noticeable stretching compared to a circle. Knowing the eccentricity helps you understand the shape and geometry of the ellipse.

Semi-major axis

The semi-major axis is half of the longest diameter of the ellipse. It measures from the center of the ellipse to the farthest point on the perimeter along the major axis. This axis is crucial as it helps define the overall size and shape of the ellipse.

In the context of orbits, the semi-major axis can tell us how far an object travels from the center point (like a planet from the sun). In our given problem, the semi-major axis was determined to be 3 units long using the eccentricity and the focus distance.

Calculating the semi-major axis is straightforward using the formula:

• \( a = \frac{c}{e} \)

Where \( c \) is the distance to a focus and \( e \) is the eccentricity, this shows the direct relationship between these elements in defining the ellipse's dimensions.

Equation of Ellipse

The equation of an ellipse depends on the orientation of its major axis and the lengths of its axes. A standard ellipse centered at the origin has either a horizontal or vertical major axis.

For a horizontal major axis:

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

For a vertical major axis:

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

Here, \( a \) and \( b \) are the semi-major and semi-minor axes, respectively. The key is to substitute the correct values based on orientation and calculate \( a \) and \( b \) beforehand to derive the specific equation.

For example, with \( a = 3 \) and \( b = \sqrt{5} \), and the major axis being vertical, the equation becomes:

• \( \frac{x^2}{5} + \frac{y^2}{9} = 1 \)

This equation captures the balance between the axes lengths and defines the set of points constituting the ellipse.