Problem 36 - Precalculus Mathematics for Calculus

Question

Find an equation for the ellipse that satisfies the given conditions. Length of major axis: \(6,\) length of minor axis: \(4,\) foci on \(x\) -axis

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{9} + \frac{y^2}{4} = 1\).

Step 1: Identify the center and orientation

Since the foci are on the \(x\)-axis, the ellipse is horizontally oriented. The center of the ellipse is typically at the origin, \((0,0)\), unless otherwise specified.

Step 2: Determine the semi-major and semi-minor axes lengths

Given that the length of the major axis is 6, the semi-major axis \(a\) is half of the major axis, so \(a = \frac{6}{2} = 3\). Similarly, since the length of the minor axis is 4, the semi-minor axis \(b\) is \(b = \frac{4}{2} = 2\).

Step 3: Calculate the distance to foci

The distance \(c\) from the center to each focus can be found using the relationship \(c^2 = a^2 - b^2\). Substituting from previous steps, \(c^2 = 3^2 - 2^2 = 9 - 4 = 5\). Therefore, \(c = \sqrt{5}\).

Step 4: Write the equation of the ellipse

The standard equation for a horizontally oriented ellipse with center at the origin is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Substituting \(a = 3\) and \(b = 2\) into the equation, we have \(\frac{x^2}{9} + \frac{y^2}{4} = 1\).

Key Concepts

Major Axis

An ellipse's major axis is its longest diameter. You can think of it as the stretched part of an oval. Often, the major axis is horizontal, but it can be vertical too, depending on how the ellipse sits in your coordinate plane. This axis runs through the center from one end to the other, passing through both foci.

• For example, in this exercise, the major axis is horizontal, and its length is 6 units.
• The center of the ellipse is at \( (0, 0) \).

The major axis isn't only a measure of spread; it carries important properties for constructing the ellipse's equation as well.

Minor Axis

The minor axis of an ellipse is like the little brother to the major axis. It is the shortest diameter that also passes through the center of the ellipse but cuts through in such a way that it is perpendicular to the major axis.

• In this example, the minor axis has a length of 4 units.
• It's placed vertically since the major axis is horizontal.

Even though it's shorter than the major axis, the minor axis plays a significant role in calculating aspects like the ellipse's equation and shape.

Foci

Foci are two special points located inside every ellipse which are crucial in defining its shape. They are always positioned on the major axis and help determine how "stretched" an ellipse is.

• The sum of the distances from any point on the ellipse to each focus is constant.
• In this case, the foci are placed along the x-axis because our major axis is horizontal.

To find the exact position of these foci, you'd use the formula \( c^2 = a^2 - b^2 \), as was done in the original problem.

Semi-Major Axis

The semi-major axis is simply half of the major axis, just as the name suggests. It runs from the center of the ellipse to one end along the major axis.

• In this exercise, the semi-major axis length is calculated as \( a = rac{6}{2} = 3 \).
• Like the major axis, it helps to define the ellipse's equation and determine foci positions.

Knowing the semi-major axis helps in deriving properties and equations since it's used directly in the standard equation of the ellipse.

Semi-Minor Axis

Much like the semi-major axis, the semi-minor axis is half of the minor axis. It stretches from the center to the edge of the ellipse along the minor axis.

• Here, the semi-minor axis length is \( b = rac{4}{2} = 2 \).
• This measurement impacts the ellipse's shape in a similar way as the semi-major axis.

In the formula for the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the semi-minor axis provides the value for \( b^2 \), further indicating its importance.