Problem 36 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. Endpoints of major axis: \((2,2)\) and \((8,2)\) Endpoints of minor axis: \((5,3)\) and \((5,1)\)

Step-by-Step Solution

Answer

The equation for the ellipse in standard form is \(\frac{(x-5)^2}{9} + \frac{(y-2)^2}{1} = 1\)

Step 1: Determine the center

Find the midpoint of the endpoints of the major axis to determine the center of the ellipse. The midpoint formula is \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\), which results in the center point \((\frac{2+8}{2}, \frac{2+2}{2}) = (5, 2)\)

Step 2: Determine the length of the semi axes

Calculate the lengths of the major and minor semi-axes. The semi major axis is half the distance between the endpoints of the major axis: \(\frac{d((8, 2), (2, 2))}{2} = \frac{\sqrt{(8-2)^2 + (2-2)^2}}{2} = 3\).The length of the semi-minor axis is half the distance between the endpoints of the minor axis: \(\frac{d((5, 3), (5, 1))}{2} = \frac{\sqrt{(5-5)^2 + (3-1)^2}}{2} = 1\)

Step 3: Write the equation in standard form

Using the formula for the equation of an ellipse in standard form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) where \((h, k)\) is the center and \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes (respectively), we substitute our values to get the equation \(\frac{(x-5)^2}{9} + \frac{(y-2)^2}{1} = 1\)

Key Concepts

Ellipse Center

In an ellipse, the center is the point equidistant from all its sides. It acts as the focal point from which the shape of the ellipse is naturally symmetrical. To find this center in a problem, particularly when given the endpoints of the major axis, use the midpoint formula.

The midpoint formula calculates the average of the x-coordinates and y-coordinates separately. For example, with endpoints \(2,2\) and \(8,2\), it looks like this:

• Calculate the average of the x-coordinates: \(\frac{2+8}{2} = 5\)
• For the y-coordinates: \(\frac{2+2}{2} = 2\)

The result is the center of the ellipse, which, in this scenario, is \(5,2\). This central point helps guide the positioning of both the major and minor axes.

Major Axis

The major axis of an ellipse is its longest diameter. It stretches across the widest part of the ellipse, passing through the center. Knowing the major axis is crucial because it defines the general shape and size of the ellipse.

To find the major semi-axis length:

• Calculate the distance between the endpoints of the major axis. Consider our example, which has endpoints \(2,2\) and \(8,2\).
• Using the distance formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\), compute the full length.
• Here, it simplifies to \(\sqrt{(8-2)^2} = 6\).
• The semi-major axis is half of this value, hence \(3\).

The full length of the major axis in the example is \(6\), with the semi-major one being \(3\).

Minor Axis

The minor axis of an ellipse is its shortest diameter. It also passes through the center, intersecting the major axis at a right angle. It's paramount to understand this as it affects the ellipse's height.

Here's how to figure out the semi-minor axis length:

• If endpoints of the minor axis are provided, as with \(5,3\) and \(5,1\), use those points.
• Apply the distance formula, much like with the major axis, but now calculate between minor axis endpoints: \(\sqrt{(5-5)^2 + (3-1)^2} = \sqrt{4} = 2\).
• The semi-minor axis then measures \(1\) when halved.

Thus, for our exercise, the full minor axis is a length of \(2\), while the semi-minor axis stands at \(1\).

Midpoint Formula

The midpoint formula is a fundamental mathematical tool when dealing with ellipses. It helps determine the center by finding the middle point between two given coordinates. This is vital in the context of ellipses where locating the center accurately is key to defining the shape.

The formula itself is simple:

• \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)

In practical application, simply add together the respective coordinates of your two points, and then divide each sum by two.

For instance, if your given points are \(2,2\) and \(8,2\), plug these into the formula:

• For x-coordinate: \(\frac{2+8}{2} = 5\)
• For y-coordinate: \(\frac{2+2}{2} = 2\)

The outcome provides the center of the ellipse, specifically \(5,2\) in this case, exemplifying the power and simplicity of the midpoint formula.