Problem 21 - Precalculus with Calculus Previews

Question

In Problems 21-40, find an equation of the ellipse that satisfies the given conditions. Vertices \((\pm 5,0),\) foci (±3,0)

Step-by-Step Solution

Answer

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

Step 1: Identify the Equation Form

The vertices and foci are on the x-axis, indicating the ellipse is horizontal. Therefore, the standard form of the equation 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 a co-vertex.

Step 2: Determine the Center

The ellipse is symmetric about the origin, so the center is at \((0,0)\). Therefore, the equation simplifies to \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

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

The vertices are given as \((\pm 5, 0)\), so the length of the semi-major axis, \(a\), is 5.

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

From the foci \((\pm 3, 0)\), we know \(c\), the distance from the center to a focus, is 3. Using the relationship \(c^2 = a^2 - b^2\), we find \(b^2 = a^2 - c^2 = 5^2 - 3^2 = 25 - 9 = 16\). Thus, \(b = 4\).

Step 5: Write the Equation

Substitute \(a^2 = 25\) and \(b^2 = 16\) into the simplified equation form: \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \).

Key Concepts

Semi-Major Axis

In an ellipse, the semi-major axis is the longest radius extending from the center to the edge of the ellipse. It defines how far the ellipse stretches along its longest dimension.
For any horizontal ellipse like the one in the exercise, the semi-major axis runs parallel to the x-axis.
The formula used here is represented by \(a\), which is half the length of the longest axis called the major axis.

• In our given problem, the vertices of the ellipse are \((\pm 5, 0)\), which means the full length of the major axis is 10 units (calculated as the distance between two vertices).
• Thus, the semi-major axis is half of that, which is 5 units.
• This makes \(a = 5\).

Understanding the semi-major axis is crucial, as it helps to determine the shape and size of the ellipse.

Semi-Minor Axis

The semi-minor axis of an ellipse is the shortest radius, extending from the center to the ellipse's edge.
It's perpendicular to the semi-major axis. For a horizontal ellipse, it runs along the y-axis.

• In our equation, \(b\) denotes the length of the semi-minor axis.
• Using the relationship \(c^2 = a^2 - b^2\), where \(c\) is the distance from the center to a focus (3 units in our example), we calculate \(b^2\).
• We know \(a = 5\) from the previous section, hence \(a^2 = 25\).
• Given \(c = 3\), \(c^2 = 9\).
• By rearranging the equation to find \(b^2\), we get \(b^2 = 25 - 9 = 16\), thus \(b = 4\).

The semi-minor axis helps define the ellipse’s compactness and symmetry.

Foci

An ellipse has two special points known as foci. They are crucial for its basic definition and play a significant role in forming the ellipse's unique shape.
Each focus is located inside the ellipse.

• For our ellipse, the foci are positioned at \((\pm 3, 0)\). This indicates that the foci are along the x-axis, making this a horizontal ellipse.
• The distance from the center of the ellipse to either focus is denoted by \(c\).
• We already determined that \(c = 3\) in our solution.
• The special property of ellipses is that the sum of the distances from any point on the ellipse to each focus is constant.

Understanding the foci helps us grasp why ellipses have this smooth oval shape and how it relates to distances from these points.

Vertices

Vertices are the endpoints of the semi-major axis. They are the points where the ellipse is widest.
For horizontal ellipses, these lie along the x-axis, while in vertical ellipses, they align along the y-axis.

• In our problem, the vertices are given as \(\pm 5, 0\).
• This confirms that the ellipse is centered at the origin with \(h, k) = (0, 0)\), stretching 5 units in each direction along the x-axis.
• The vertices signify the farthest boundaries on the ellipse's horizontal stretch.

Vertices are vital for establishing the ellipse's orientation and overall reach across its major axis.