Problem 39 - Precalculus: Mathematics for Calculus

Question

Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. Foci: \((\pm 4,0),\) vertices: \((\pm 5,0)\)

Step-by-Step Solution

Answer

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

Step 1: Identify Components of the Ellipse

To find the equation of the ellipse, first note the information given: Foci are \((\pm 4,0)\) and vertices are \((\pm 5,0)\). These points lie on the x-axis, indicating a horizontal major axis.

Step 2: Use the Standard Form of an Ellipse Equation

For an ellipse with a horizontal major axis, the standard form is \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \], where \(h, k\) is the center, \(a\) is the distance from the center to a vertex, and \(b\) is the distance from the center to the endpoint of the minor axis. Here, the center is \( (0,0) \), since the foci and vertices are symmetric about the origin.

Step 3: Determine 'a' and 'c'

The distance from the center to a vertex, \(a\), is 5 since the vertices are \(\pm 5, 0\). The distance from the center to a focus, \(c\), is 4 as the foci are \(\pm 4, 0\).

Step 4: Calculate 'b' Using Relationship between 'a', 'b', and 'c'

For ellipses, \(c^2 = a^2 - b^2\). Substitute the known values: \(4^2 = 5^2 - b^2\). Hence, \(16 = 25 - b^2\) leading to \(b^2 = 9\). Thus, \(b = 3\).

Step 5: Write the Equation of the Ellipse

Substitute the values of \(a\) and \(b\) into the standard form: \[ \frac{x^2}{5^2} + \frac{y^2}{3^2} = 1 \]. Simplify to: \[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \].

Key Concepts

Foci of an Ellipse

The foci of an ellipse are two fixed points located inside the ellipse. Their position is such that the sum of the distances from any point on the ellipse to these two foci is constant. In this exercise, the foci are at \( (\pm 4,0) \). This tells us crucial information about the ellipse, especially how stretched it is along the major axis.
For an ellipse centered at the origin, if the foci are located on the x-axis, it indicates that the ellipse is horizontally oriented. The distance from the center of the ellipse to each focus is denoted as \( c\). In the given problem, the distance \( c = 4\). This distance will help us find other properties of the ellipse, like the lengths of the axes.

Vertices of an Ellipse

Vertices are the points where the ellipse is widest. They lie on the major axis, which is the longer of the two axes of the ellipse. In this case, the vertices are given at \( (\pm 5,0) \). This tells us that the ellipse stretches 5 units from the center along the x-axis.
The distance from the center to each vertex is denoted as \( a\). Here, \( a = 5\). This is vital for finding the ellipse's equation as it defines the semi-major axis length. Knowing the vertices also helps determine the orientation of the ellipse, which is horizontal in this case.

Standard Form of Ellipse

To express the ellipse mathematically, we use the standard form of the ellipse equation. For an ellipse centered at \((h, k)\) with a horizontal major axis, the standard form is \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \].
If the ellipse is centered at the origin, then \(h = 0\) and \(k = 0\). Therefore, our equation simplifies to \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \].
Given that \( a = 5 \) and \( b^2 = 9 \), substituting these into the formula, we derive the ellipse equation: \[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]. This highlights how far the ellipse stretches along its axes.

Major Axis of an Ellipse

The major axis of an ellipse is its longest diameter. It passes through both the center and the foci. The length of the major axis is twice the distance \( a\). In this exercise, since the vertices are located at \( (\pm 5, 0) \), the length of the major axis is \( 2a = 10\).
This axis defines the overall direction of the longest part of the ellipse, which in this problem is horizontal. The major axis is crucial as it dictates how stretched the ellipse is along one direction and significantly impacts its shape and the calculations made to determine the ellipse's equation.