Problem 40 - Precalculus: Mathematics for Calculus

Question

Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. Foci: \((0, \pm 3),\) vertices: \((0, \pm 5)\)

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \).

Step 1: Identify the Center

For the given ellipse, the foci are at (0, ±3) and the vertices are at (0, ±5). Both foci and vertices are centered around the origin (0,0), which indicates that the center of the ellipse is at (0,0).

Step 2: Determine the Orientation

The vertices and foci are aligned along the y-axis. This means the major axis is vertical.

Step 3: Identify Lengths of Major and Minor Axes

Since the vertices are at (0, ±5), the length of the semi-major axis (a) is 5. The distance from the center to either focus is 3, so the length of the semi-minor axis (b) must be found using the relationship between a, b, and the distance to the foci (c): a^2 = b^2 + c^2.

Step 4: Apply the Relationship a^2 = b^2 + c^2

We know a = 5 and c = 3. Use the relationship to find b: 5^2 = b^2 + 3^2. This simplifies to 25 = b^2 + 9.

Step 5: Solve for b^2

Subtract 9 from both sides to solve for b^2: b^2 = 25 - 9 = 16. Thus, b = 4.

Step 6: Write the Equation of the Ellipse

For a vertically oriented ellipse with center at (0,0), the equation is given by \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), where a is the length of the semi-major axis and b is the length of the semi-minor axis. Substitute a = 5 and b = 4 into this equation: \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \).

Key Concepts

Ellipse Foci

Foci are two special points on an ellipse that have an important geometric property. An ellipse has two foci (singular is focus), which help in defining the curve. The sum of the distances from any point on the ellipse to these two foci is constant.

For the specific ellipse given in our exercise, the foci are located at

• (0, 3)
• (0, -3)

This tells us that the foci are positioned vertically. The center is at the origin (0,0). In any ellipse around the center, the foci are symmetric with respect to the center. This characteristic symmetry centers everything around the origin and aligns with the axes.

The distance from the center to either focus, denoted as "c," is an essential measurement, resulting in the equation relating to an ellipse's axes: The larger the distance between the foci, the more elongated the ellipse becomes.

Ellipse Vertices

Vertices are key points on an ellipse, marking the endpoints of the major (longest) axis. In a vertically oriented ellipse, such as the one described in the exercise, the vertices lie directly above and below the center.

For the ellipse in our example, the vertices are located at:

• (0, 5)
• (0, -5)

These points signify that the major axis is vertical, stretching upwards and downwards. The vertices determine half of the total major axis length, known as the semi-major axis, represented by "a."

The distance from the center to a vertex helps determine the ellipse's shape, influencing one component of its guiding equation, especially since "a" is crucial in calculating the equation.

Semi-Major Axis

The semi-major axis is a fundamental part of an ellipse, representing half the longest diameter. It dictates how elongated or wide the ellipse appears and always runs through the foci.

In our given exercise, since the vertices are at

• (0, 5)
• (0, -5)

The semi-major axis, "a," measures 5. This axis extends vertically along the y-axis, which is consistent with the position of the vertices.

Understanding the semi-major axis is essential because it is used in structuring the standard ellipse equation, given as \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \] where "a" represents the square of the semi-major axis.
As seen, the semi-major axis helps define the axis of symmetry and structure of the ellipse.