Problem 28 - Algebra 2

Question

Write an equation for the ellipse that satisfies each set of conditions. endpoints of minor axis at (0, 5) and (0, -5), foci at (12, 0) and (-12, 0)

Step-by-Step Solution

Answer

The equation is \(\frac{x^2}{169} + \frac{y^2}{25} = 1\).

Step 1: Understand the Structure of an Ellipse

An ellipse has two main axes: the major axis and the minor axis. The equation of an ellipse in standard form centered at the origin is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) where \(a\) represents the semi-major axis and \(b\) the semi-minor axis if \(a \geq b\). If the ellipse is oriented along the x-axis, \(a > b\).

Step 2: Determine the Length of the Minor Axis

The given endpoints of the minor axis are (0, 5) and (0, -5). This indicates that the full length of the minor axis is 10, as the y-coordinates range from -5 to 5. Hence, the semi-minor axis, \(b\), is 5.

Step 3: Identify the Foci Distance and Determine Major Axis

The foci are given at (12, 0) and (-12, 0). The distance of the foci from the center is represented by \(c = 12\). Here, \(c > b\), indicating that the major axis is along the x-axis. The relationship between the axes in an ellipse is \(c^2 = a^2 - b^2\).

Step 4: Calculate the Length of the Major Axis

Use the equation \(c^2 = a^2 - b^2\). We know \(c = 12\) and \(b = 5\). Substitute these into the equation to find \(a\):\[12^2 = a^2 - 5^2\]\[144 = a^2 - 25\]\[a^2 = 169\]\[a = 13\]. Hence, the semi-major axis \(a\) is 13.

Step 5: Write the Equation of the Ellipse

Now that we know \(a = 13\) and \(b = 5\), substitute \(a^2\) and \(b^2\) into the standard form of the ellipse equation:\[\frac{x^2}{13^2} + \frac{y^2}{5^2} = 1\]Which simplifies to:\[\frac{x^2}{169} + \frac{y^2}{25} = 1\]

Key Concepts

Major Axis

The major axis of an ellipse is the longest diameter passing through its center. It is aligned with the largest value between the lengths of the semi-major and semi-minor axes. In the context of our exercise, the major axis is determined by the foci given at

• (12, 0)
• (-12, 0)

This implies that the major axis is aligned along the x-axis. The total length of the major axis is twice the semi-major axis, denoted as 26, or more simply said, it’s calculated as the sum of

1. 13 units to each side from the origin, 0.

Minor Axis

The minor axis is the shortest diameter of the ellipse, perpendicular to the major axis. It is critical for defining the shape of the ellipse. In the provided problem, the endpoints of the minor axis are given as

• (0, 5)
• (0, -5).

This shows that the minor axis is vertically aligned. The total length of the minor axis can be derived by calculating the distance between these endpoints, resulting in 10 units. Therefore, the minor axis provides an essential constraint to derive other parameters of the ellipse.

Foci of an Ellipse

Foci are two fixed points on the interior of an ellipse that help define its shape. The sum of distances from any point on the ellipse to the foci is constant. In this scenario, the coordinates for the foci are

• (12, 0)
• (-12, 0)

This provides a significant insight that the ellipse extends further along the x-axis than the y-axis, thus confirming the position of the major axis. In ellipses, the distance between the center and either foci is denoted by the term \(c\), which has been given as 12 in this case. This is a fundamental concept necessary to determine the various axes lengths using key equations that relate to ellipse measurements.

Standard Form of an Ellipse

The standard form of an ellipse's equation helps describe all its components in a solvable format. When centered at the origin, it is expressed as \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Here, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis. For the given exercise, we substitute in the values obtained for \(a\) and \(b\), being 13 and 5 respectively, into this formula. This allows us to write \[\frac{x^2}{169} + \frac{y^2}{25} = 1\] which completely characterizes the ellipse based on the given conditions.

Semi-Major Axis

The semi-major axis represents half the length of the major axis, highlighting its importance in defining the size of an ellipse. In mathematical terms, it's one part of the equation that

1. determines the major axis's length
2. influences the ellipse’s stretch factor.

From the exercise, we calculate the semi-major axis using the distance from the center to a focus point along the x-axis (\(a = \sqrt{169} = 13\)), which confirms its configuration centered at the origin and governing the elliptical stretch along the x-axis.

Semi-Minor Axis

The semi-minor axis is crucial for understanding the shape and scale of an ellipse, representing half the minor axis' length. In our exercise, the semi-minor axis is deduced to be 5, stemming from the minor axis endpoints,

• (0, 5)
• (0, -5)

Through analyzing the distance along the y-axis, we ascertain this relationship. This parameter

1. determines how compact or wide an ellipse appears vertically.

The minor axis directly influences the horizontal spread when put into the standard ellipse formula, cementing its role alongside the semi-major axis.