Problem 3 - Algebra 2

Question

Write an equation for the ellipse that satisfies each set of conditions. Write an equation for the ellipse that satisfies each set of conditions. endpoints of major axis at (0,10) and \((0,-10),\) foci at (0,8) and (0,-8)

Step-by-Step Solution

Answer

Equation: \( \frac{x^2}{36} + \frac{y^2}{100} = 1 \).

Step 1: Identify the Center

To find the center of the ellipse, we take the midpoint of the major axis endpoints. Since the major axis endpoints are (0,10) and (0,-10), the center is at the midpoint \( (0, 0) \).

Step 2: Determine the Major and Minor Axes Lengths

The major axis endpoints give the length of the major axis. The distance from (0,10) to (0,-10) is 20, hence the length of the semi-major axis is \( a = 10 \). The coordinates of the foci (0,8) and (0,-8) suggest a focus distance \( c = 8 \).

Step 3: Use the Relationship Between the Semi-Major, Semi-Minor Axes, and Foci

For ellipses, the relationship between semi-major axis \( a \), semi-minor axis \( b \), and the distance to the foci \( c \) is given by the equation \( c^2 = a^2 - b^2 \). Here, \( c=8 \) and \( a=10 \), so: \[ 8^2 = 10^2 - b^2 \] \( 64 = 100 - b^2 \). Solving for \( b \), we find \( b^2 = 36 \), thus \( b = 6 \).

Step 4: Write the Equation of the Ellipse

We now have \( a = 10 \) and \( b = 6 \). The standard equation of an ellipse centered at \( (0,0) \) with its major axis along the y-axis is: \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \] Substitute \( b = 6 \) and \( a = 10 \) into the equation to get: \[ \frac{x^2}{36} + \frac{y^2}{100} = 1 \].

Key Concepts

Major and Minor Axes

In an ellipse, the major axis is the longest diameter, stretching across the widest part of the shape. It passes through the center, foci, and vertices of the ellipse. For vertical ellipses, this means that the major axis runs top to bottom, while for horizontal ellipses, the axis would run left to right. In this example, the major axis is clearly vertical as it stretches from

• (0, 10) to (0, -10)

marking a major axis length of 20. That means each end of the major axis reaches 10 units from the center on either side. The minor axis is perpendicular to the major axis. It is the shorter diameter of the ellipse, positioned at the widest part perpendicular to the major axis. For our ellipse centered at (0, 0), it is horizontal.

Foci of an Ellipse

The foci (plural for focus) are two special points located along the major axis, equidistant from the center, that define the curvature of the ellipse. The sum of the distances from any point on the ellipse to the two foci is constant. In the problem provided, the foci are located at:

• (0, 8) and (0, -8)

which means they are 8 units away from the origin along the y-axis. Having foci further away from the center makes the ellipse appear more elongated, while closer foci produce a more circular shape. In this case, the foci help define the shape along with the semi-major axis.

Center of an Ellipse

The center of an ellipse is the midpoint of both the major and minor axes. It serves as a reference point from which the distances to various elements of the ellipse, like the vertices and foci, are measured. In the provided exercise, the center is calculated as the midpoint of the major axis endpoints:

• (0, 10) and (0, -10)

This yields a center at (0, 0). It’s important as the equation of the ellipse is often represented relative to this central point. In our specific example, the equation takes a familiar standard form with the center at the origin.

Semi-Major Axis

A semi-major axis is half of the major axis and extends from the center to a vertex. It is one of the most defining dimensions of an ellipse. For our exercise, the major axis is 20 units long. Therefore, the semi-major axis is

• 10 units

in either direction from the center along the y-axis. This means that from the center (0, 0), it reaches up to the points (0, 10) and (0, -10) along the vertical axis. The length of the semi-major axis is critical in determining the stretch and shape of the ellipse and it features prominently in the standard ellipse equation \[\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\]where a is the semi-major axis length.

Semi-Minor Axis

The semi-minor axis is half the length of the minor axis; it extends from the center to the ellipse’s edge perpendicular to the major axis. In this exercise, after using the foci and semi-major axis, we found that the semi-minor axis length is:

• 6 units

This length forms the width of the ellipse when referenced from the center. The semi-minor axis, denoted as "b", is crucial in forming the ellipse along with the length of the semi-major axis. The relationship between the ellipses axes and its foci is captured in the equation \[c^2 = a^2 - b^2\]where

• c is the distance from the center to a focus

and helps find dimensions crucial to drawing the ellipse correctly. In our example, it contributes to determining how elongated the vertical shape is relative to its width.