Problem 31 - A Graphical Approach to College Algebra

Question

Find an equation for each ellipse. Center \((3,-2) ; a=5 ; c=3 ;\) major axis vertical

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{(x-3)^2}{16} + \frac{(y+2)^2}{25} = 1 \).

Step 1: Understand the Problem

We're given an ellipse with a specific center (\((3, -2)\)), a given semi-major axis length (\(a=5\)), and a given distance from the center to the focus (\(c=3\)). Our task is to find the equation of this ellipse, knowing that the major axis is vertical.

Step 2: Determine the Form of the Equation

Since the major axis is vertical, the general form of the ellipse's equation will be: \[ \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \]where \((h, k)\) is the center of the ellipse, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.

Step 3: Calculate \(b\) Using the Relationship

The relationship between \(a\), \(b\), and \(c\) in ellipses is given by the equation \(c^2 = a^2 - b^2\).Plugging in the values:\[c = 3, \quad a = 5 \]\[3^2 = 5^2 - b^2\]\[9 = 25 - b^2\]\[b^2 = 16\]\[b = 4\]

Step 4: Substitute Values into the Ellipse Equation

Now that we have \(b = 4\), and we know the center \((h, k) = (3, -2)\), we substitute these values back into the general equation:\[ \frac{(x-3)^2}{16} + \frac{(y+2)^2}{25} = 1 \]This is the equation of the ellipse with the given parameters.

Key Concepts

Equation of an Ellipse

The equation of an ellipse is a mathematical expression that defines the set of all points satisfying particular conditions related to distances. An ellipse has two standard forms, depending on the orientation of its major axis, either horizontal or vertical. If the major axis is vertical, as in the problem's example, the equation is:
\[ \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \]

• \((h, k)\) represents the center of the ellipse.
• \(a\) is the length of the semi-major axis.
• \(b\) is the length of the semi-minor axis.

The concept can be thought of as an adaptation of the circle's equation, where the circle is a special case of an ellipse with equal axes. The denominators \(a^2\) and \(b^2\) scale the ellipse in the respective directions, ensuring that it crosses axes at the right lengths.

Major and Minor Axes

In an ellipse, the major and minor axes refer to the longest and shortest diameters, respectively. Here's how they are defined:

• **Major Axis**: The longest line that runs through the center of the ellipse and its foci. In the given problem, since the major axis is vertical, it stretches vertically from one end of the ellipse to the other, crossing through both foci and the center.
• **Minor Axis**: The shortest line through the center that is perpendicular to the major axis. It's also symmetric about the center.

The lengths of these axes are key in determining the ellipse's equation. The major axis length is twice the semi-major axis \(2a\), and the minor axis length is twice the semi-minor axis \(2b\). The relationship between these lengths helps to identify whether the ellipse is stretched more in one direction compared to the other.

Center of Ellipse

The center of an ellipse is a pivotal point in its geometry. It is the midpoint of both axes:

• It acts as a symmetry point, meaning each half of the ellipse is a mirror image of the other half.
• In the equation of the ellipse \((h, k)\), \(h\) and \(k\) denote the \(x\) and \(y\) coordinates of the center, respectively.

For the given problem, the center is located at \((3, -2)\). This implies that all calculations involving \(h\) and \(k\) use these values. Adjusting for the center shifts the entire ellipse from the origin to the specified coordinates, ensuring it sits properly on the coordinate plane.

Focus of Ellipse

The focus (plural: foci) of an ellipse is a unique aspect of its structure. These are two points located along the major axis, equidistant from the center. The foci serve as a "gravity center", where the sum of the distances from any point on the ellipse to the foci is constant. Here's more about them:

• The distance from the center to each focus is noted as \(c\).
• In any ellipse, the relationship between \(a\), \(b\), and \(c\) is given by \(c^2 = a^2 - b^2\).

In the problem, \(c = 3\), indicating that each focus is 3 units away from the center along the major axis. Understanding foci is crucial because, although they aren't part of the main equation, they help explain the ellipse's geometry and its property of maintaining total distance from the foci.