Problem 40 - Precalculus Mathematics for Calculus

Question

Find an equation for the ellipse that satisfies the given conditions. Endpoints of minor axis: \((0, \pm 3),\) distance between foci: 8

Step-by-Step Solution

Answer

The equation is \(\frac{x^2}{25} + \frac{y^2}{9} = 1\).

Step 1: Identify the Key Characteristics of the Ellipse

An ellipse is defined by its center, major axis, minor axis, and distance between its foci. Given the endpoints of the minor axis are \((0, \pm 3)\), the center of the ellipse is at point \((0, 0)\) and the minor axis length is \(6\) because the distance between \((0, 3)\) and \((0, -3)\) is 6. The distance between the foci is given as \(8\). The distance \(c\) between each focus from the center is \(\frac{8}{2} = 4\).

Step 2: Determine the Length of the Major Axis

The standard equation of an ellipse with the center at the origin \((0, 0)\) is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.Since \(b^2 = 9\) because the minor axis length is 6, we can use the relationship \(c^2 = a^2 - b^2\) (from the fundamental ellipse property) to find \(a\). We know \(c = 4\), so \(c^2 = 16\). Therefore, \(16 = a^2 - 9\).

Step 3: Calculate the Value of the Major Axis

From \(16 = a^2 - 9\), we solve for \(a^2\): \[a^2 = 16 + 9 = 25\]Thus, the semi-major axis \(a\) has length \( \sqrt{25} = 5\). This confirms that the major axis length is 10.

Step 4: Write the Equation of the Ellipse

Now that we have the lengths of the semi-major and semi-minor axes, we know the equation is\[\frac{x^2}{25} + \frac{y^2}{9} = 1\]This equation represents the ellipse centered at the origin with a major axis length of \(10\) (horizontal axis) and a minor axis length of \(6\).

Key Concepts

Major Axis

The major axis of an ellipse is the longest diameter and it spans the entire length of the ellipse. It runs through the center and touches both vertices, giving the ellipse its elongated shape.
In terms of our specific exercise, understanding the major axis is crucial, as it helps us set up the ellipse equation correctly.When the endpoints of the minor axis are given and the distance between foci are known, the major axis is often horizontal. This is derived from the fact that the longer axis needs this orientation to house the foci and other characteristics within the ellipse.To calculate the length of the major axis of an ellipse:

• Identify the value of the semi-major axis, denoted as \(a\).
• The major axis length is \(2a\).

In our example, since the semi-major axis \(a\) is 5, the total length of the major axis is \(2 \times 5 = 10\). This horizontal span is essential since it encompasses the foci and aids in defining the whole geometry of the ellipse.

Minor Axis

The minor axis is the shortest diameter of an ellipse and is perpendicular to the major axis. It also passes through the center of the ellipse.In many exercises similar to ours, the endpoints of the minor axis are among the first pieces of data available. Knowing these endpoints helps in determining the center of the ellipse easily since it lies at the midpoint between these ends.
In this specific problem:

• The endpoints of the minor axis provided are \((0, \pm 3)\).
• The calculation for the minor axis is simply finding the distance between these points, which totals 6.
• It determines \(b\), the semi-minor axis, which is half this length, or \(3\).

The minor axis therefore measures a total length of 6 in our exercise, which gives the vertical stretch of the ellipse, balancing the horizontal expansion provided by the major axis.

Distance Between Foci

The distance between the foci of an ellipse is an important feature as it impacts the overall shape of the ellipse. Foci (plural of focus) are two special points located inside the ellipse, along the major axis.This concept is key in distinguishing ellipses from circles. In an ellipse, this distance is non-zero, while for circles, the foci converge at the circle's center.In our exercise, the given distance directly influences how we determine the major axis. Specifically:

• The total distance from one focus to the other is provided as 8.
• Since this is \(2c\) (where \(c\) is the distance from the center to a single focus), \(c = 4\).

The relationship between \(a\), \(b\), and \(c\) in an ellipse is expressed through the equation \(c^2 = a^2 - b^2\). Knowing \(c\) allows us to solve for \(a\) once \(b\) is known, thus enabling us to find the semi-major axis and fully define our ellipse.