Problem 24 - Precalculus with Calculus Previews

Question

Find an equation of the ellipse that satisfies the given conditions. Vertices \((0,\pm 7),\) foci (0,±3)

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{x^2}{40} + \frac{y^2}{49} = 1 \).

Step 1: Identify the Orientation of the Ellipse

The vertices of the ellipse are given as \(0, \pm 7\), and the foci are \(0, \pm 3\). Since both vertices and foci lie along the y-axis, the major axis is vertical.

Step 2: Write the General Equation of the Vertical Ellipse

For a vertical orientation, the equation for an ellipse is \[\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\]where \(a\) represents the semi-major axis, and \(b\) is the semi-minor axis.

Step 3: Determine the Length of the Semi-Major Axis

The vertices \(0, \pm 7\) indicate that the length of the semi-major axis is \(a = 7\). Thus, \(a^2 = 49\).

Step 4: Determine the Length of the Semi-Minor Axis Using the Foci

The foci \(0, \pm 3\) provide the value of \(c = 3\). In an ellipse, \(c^2 = a^2 - b^2\). Substitute the known values:\[3^2 = 7^2 - b^2\]\[9 = 49 - b^2\]\[b^2 = 40\]

Step 5: Plug Values into the General Equation of the Ellipse

Now that we know \(a^2 = 49\) and \(b^2 = 40\), substitute these into the equation:\[\frac{x^2}{40} + \frac{y^2}{49} = 1\]

Key Concepts

Vertical Ellipse

An ellipse with a vertical orientation is called a vertical ellipse. Understanding its characteristics is crucial when determining the equation of such an ellipse. This type of ellipse has its major axis aligned parallel to the y-axis.

• In a vertical ellipse, the largest distance across the shape, or the length of the major axis, runs vertically.
• The vertices (the furthest points on the ellipse from the center) are located on the line parallel to the y-axis.

In our example, the vertices given as (0, ± 7) clearly indicate that the major axis is aligned vertically. Hence, we conclude that this is a vertical ellipse.

Semi-Major Axis

The semi-major axis is one of the most important components in understanding an ellipse. It represents half the length of the major axis, which is the longest line segment that can be drawn through the center.

• For a vertical ellipse, the semi-major axis extends from the center to a vertex along the y-axis.
• The length of the semi-major axis is denoted as 'a'.

In the problem presented, the semi-major axis measures 7 units, as indicated by the vertices at (0, ± 7). This gives us a value of a = 7, and consequently, a^2 = 49. Understanding this helps us position the equation correctly within the ellipse's geometry.

Semi-Minor Axis

Another crucial element of the ellipse is the semi-minor axis. This is the shorter distance across the ellipse, found perpendicular to the semi-major axis.

• For a vertical ellipse, the semi-minor axis stretches horizontally across the ellipse's center.
• The length of this axis is represented by 'b'.

To calculate this in our ellipse, we use the relationship c^2 = a^2 - b^2, where c represents the distance from the center to a focus (foci are given by the points (0, ± 3)).
Given c = 3, inserting the known values into the formula allows us to solve for b^2: 3^2 = 7^2 - b^2 results in b^2 = 40. This value will be helpful in the final equation of the ellipse.

Foci of Ellipse

The foci are unique points inside an ellipse that help in defining its shape. Every ellipse has two foci, and the sum of distances from any point on the ellipse to the foci is constant.

• For our vertical ellipse, these foci lie on the y-axis, as indicated by the coordinates (0, ± 3).
• The distance between the center of the ellipse and each focus is denoted by 'c'.

The relationship between the semi-major axis, the semi-minor axis, and the distance to the foci is given by the expression c^2 = a^2 - b^2. This relationship allows us to solve for the necessary parameters to construct the ellipse's equation. Recognizing the foci helps in visualizing the ellipse's shape and ensuring accuracy in its equation derivation.