Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, use the given information about the graph of each ellipse to determine its equation. Center (4,2)\(;\) vertex (9,2)\(;\) one focus: \((4+2 \sqrt{6}, 2)\).

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{(x-4)^2}{25} + \frac{(y-2)^2}{1} = 1 \).

Step 1: Identify the Center

The center of the ellipse is given as \(4, 2\). This means our ellipse is centered at the point \(h, k = 4, 2\).The standard form of an ellipse's equation with a horizontal major axis is: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]

Step 2: Determine the Length of the Major Axis

The vertex is given as \(9, 2\). The distance from the center \(4, 2\) to the vertex \(9, 2\) is the semi-major axis, \(a\).Calculate \(a\):\[ a = |9 - 4| = 5 \]

Step 3: Find the Distance to the Focus

The coordinates of one focus are given as \(4 + 2 \sqrt{6}, 2\). This distance from \(4, 2\) to \(4 + 2 \sqrt{6}, 2\) is the focal distance, \(c\).Calculate \(c\):\[ c = 2 \sqrt{6} \]

Step 4: Calculate the Semi-Minor Axis

Use the relationship \(c^2 = a^2 - b^2\) to find \(b\).We know: \[ a = 5, \ c = 2 \sqrt{6}, \ a^2 = 25, \ c^2 = (2 \sqrt{6})^2 = 24\] Plug these into the equation: \[ c^2 = a^2 - b^2 \implies 24 = 25 - b^2 \]Solve for \(b^2\): \[ b^2 = 25 - 24 = 1 \]

Step 5: Write the Ellipse Equation

Substitute the values of \(h, k, a^2, \ ext{and } b^2\) into the standard form of the ellipse equation:\[ \frac{(x-4)^2}{25} + \frac{(y-2)^2}{1} = 1 \]

Key Concepts

Ellipse Center

An ellipse's center is a crucial component that determines its position on a coordinate plane. In the exercise provided, the center is given at the point \(4, 2\). This information is key because it serves as the origin for the ellipse's equation, particularly the terms \(h, k\) in the standard formula.The center \(h, k\) is the midpoint from which all radial measurements, like the semi-major and semi-minor axes, are determined. This means that every aspect of the ellipse's geometry originates from this central point.
The coordinates specified, \(h = 4\) and \(k = 2\), replace \((x-h)\) and \((y-k)\) in the standard ellipse equation, rendering it specific to this ellipse:\[ \frac{(x-4)^2}{a^2} + \frac{(y-2)^2}{b^2} = 1 \]

Major Axis

The major axis of an ellipse is its longest diameter, running through both the foci and the center. It defines the ellipse's overall length, ensuring it extends symmetrically around its center.In this example, the vertex given is \(9, 2\), and the center is \(4, 2\). The major axis is horizontal due to these coordinates sharing the same y-value. The semi-major axis length, denoted as \(a\), is half the total length of the major axis and is calculated using the distance from the center to the vertex:- Calculate \(a\): \(|9 - 4| = 5\).With \(a = 5\), the full major axis length is \(2a = 10\). Recognizing the major axis is essential for understanding the scale and orientation of the ellipse.

Focal Distance

The focal distance, denoted as \(c\), is the measure from the center of the ellipse to each focus. The foci are located symmetrically on the major axis, indicating the ellipse's overall shape. Using the equation's given focus \(4 + 2 \sqrt{6}, 2\), we calculate \(c\) as follows:- \(c = 2 \sqrt{6}\).

• This result shows how far the foci are from the center along the major axis.
• The relationship between \(a\) (semi-major axis), \(b\) (semi-minor axis), and \(c\) (focal distance) is given by: \(c^2 = a^2 - b^2\).

Understanding focal distance aids in grasping the ellipse's eccentricity, which signifies how "stretched" it is.

Semi-Minor Axis

The semi-minor axis, represented by \(b\), is the shorter radius of the ellipse that extends perpendicularly to the major axis at the center. It measures half of the minor axis.To calculate \(b\), we'll use the relation \(c^2 = a^2 - b^2\) from the exercise:- Since \(a^2 = 25\) and \(c^2 = 24\):- Solving \(b^2 = 25 - 24\) yields \(b^2 = 1\).- Thus, \(b = 1\).Defining the semi-minor axis is vital because it helps describe the ellipse's narrowest span, determining how wide or slender the shape appears when compared to the major axis.