Problem 31 - Precalculus with Calculus Previews

Question

Find an equation of the ellipse that satisfies the given conditions. Foci \((\pm \sqrt{2}, 0)\), length of minor axis 6

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{11} + \frac{y^2}{9} = 1\).

Step 1: Identify the Center and the Orientation

The problem provides the foci at \((\pm \sqrt{2}, 0)\), which indicates that the ellipse is centered at the origin \((0,0)\) and is oriented along the x-axis. This suggests that the equation of the ellipse will have the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Step 2: Determine the Distance from the Center to the Foci

The distance from the center of the ellipse to each focus is given by \(c\). Since the foci are \((\pm \sqrt{2}, 0)\), \(c = \sqrt{2}\).

Step 3: Use the Information About the Minor Axis

The length of the minor axis is 6, which means \(2b = 6\). Solving this gives \(b = 3\).

Step 4: Use the Relationship Between \(a\), \(b\), and \(c\)

For an ellipse, the relationship between \(a\), \(b\), and \(c\) is given by \(c^2 = a^2 - b^2\). Substituting the known values, we have \(2 = a^2 - 3^2\).

Step 5: Solve for \(a^2\)

Solve the equation \(2 = a^2 - 9\) to find \(a^2\). Adding 9 to both sides gives \(a^2 = 11\).

Step 6: Write the Equation of the Ellipse

Using the values found, substitute \(a^2 = 11\) and \(b^2 = 9\) into the equation of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Thus, the equation is \(\frac{x^2}{11} + \frac{y^2}{9} = 1\).

Key Concepts

Foci of an Ellipse

The foci of an ellipse are two special points inside the ellipse. They have a unique property—any point on the ellipse has a total distance to the two foci that is constant. In our given problem, the foci are at

• \((-\sqrt{2}, 0)\) and
• \((\sqrt{2}, 0)\).

This means each focus is positioned symmetrically along the x-axis, around the ellipse's center at the origin \((0,0)\). Understanding the foci is essential as they also determine the ellipse's major axis, which is oriented along the distance between them.

Minor Axis of an Ellipse

The minor axis is the shortest diameter of an ellipse. It passes through the center and is perpendicular to the major axis. For the ellipse in this exercise, the length of the minor axis is given as 6 units. This tells us that from one side of the ellipse to the other, along the minor axis, measures a total of 6 units. Since it passes through the center, half of this length, which is 3 units, is the value of \(b\). This means \(b = 3\) and \(b^2 = 9\).
The minor axis is important as it helps define the scale and shape of the ellipse. While the major axis determines the longer stretch of the ellipse, the minor axis gives us the scope of its shorter spread.

Ellipse Orientation

Ellipse orientation indicates how the ellipse is stretched across the coordinate plane. In this exercise, the foci are on the x-axis, \((\pm\sqrt{2}, 0)\), which signifies that the major axis is horizontal. Therefore, the ellipse is oriented along the x-axis. This means the ellipse's wide stretch or major axis runs left and right, while the minor axis, being perpendicular, runs vertically. To represent this orientation in the standard ellipse equation format, we place the \(a^2\) term with the x-variable and \(b^2\) term with the y-variable:

• \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

This common format highlights the orientation by showing that x is associated with the larger or equal axis length.

Ellipse Properties

Ellipses have several distinctive properties that set them apart in geometry:

• The sum of the distances from any point on the ellipse to the two foci is constant.
• The major axis is the longest diameter of the ellipse, passing through the foci.
• The minor axis is the shortest diameter, perpendicular to the major axis at the center.
• The formula for the ellipse centered at \((0,0)\) is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(2a\) and \(2b\) are the lengths of the major and minor axes respectively.

In our example:

• \(a^2 = 11\)
• \(b^2 = 9\), and
• \(c = \sqrt{2} \) satisfying \(c^2 = a^2 - b^2\).

These properties not only define the shape of an ellipse, but they also provide critical insights into solving more complex problems involving ellipses.