Problem 23 - Algebra 2

Question

Find the coordinates of the center and foci and the lengths of the major and minor axes for the ellipse with the given equation. Then graph the ellipse \(27 x^{2}+9 y^{2}=81\)

Step-by-Step Solution

Answer

Center: (0, 0), Major Axis: 6, Minor Axis: 2\sqrt{3}, Foci: (0, \pm\sqrt{6}).

Step 1: Write Equation in Standard Form

To start, divide the given equation by 81 to write it in standard form. The equation is:\[ 27x^2 + 9y^2 = 81 \]Divide each term by 81:\[ \frac{27x^2}{81} + \frac{9y^2}{81} = \frac{81}{81} \]Simplified, this becomes:\[ \frac{x^2}{3} + \frac{y^2}{9} = 1 \]

Step 2: Identify Major and Minor Axes

The standard form of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) with \( b^2 > a^2 \) indicating a vertical major axis. Here, \( a^2 = 3 \) and \( b^2 = 9 \). Since \( b^2 > a^2 \), the major axis is vertical.

Step 3: Determine the Center

For an equation of the form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), the center of the ellipse is \((h, k)\). Here, \( h = 0 \) and \( k = 0 \), so the center is \((0, 0)\).

Step 4: Calculate Lengths of Major and Minor Axes

The length of the major axis is \(2b\) and the minor axis is \(2a\). Since \( a^2 = 3 \), \( a = \sqrt{3} \) and \( b^2 = 9 \), \( b = 3 \). Thus, the major axis length is \(2b = 2 \times 3 = 6\) and the minor axis length is \(2a = 2 \times \sqrt{3}\).

Step 5: Find the Foci

The foci are determined using the formula \( c^2 = b^2 - a^2 \). Here, \( c^2 = 9 - 3 = 6 \), so \( c = \sqrt{6} \). Thus, the foci are located at \( (0, c) \) and \( (0, -c) \), which are approximately at \( (0, \sqrt{6}) \) and \( (0, -\sqrt{6}) \).

Key Concepts

Center of Ellipse

The center of an ellipse is the point that is equidistant from all sides of the ellipse and serves as the midpoint of both the major and minor axes. In an ellipse equation of the form \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]\((h, k)\) are the coordinates of the center.
In this case, the equation simplifies to \[ \frac{x^2}{3} + \frac{y^2}{9} = 1 \], where \(h = 0\) and \(k = 0\).
Thus, the center of the ellipse is at the origin, \((0, 0)\). This is where you would start if you were to draw the ellipse.

• Center helps in determining the orientation of the ellipse.
• It acts as a reference point for calculating distances to the vertices and foci.

Major and Minor Axes

Ellipses are characterized by their major and minor axes, which define their shape and orientation. The major axis is the longer axis and the minor axis is the shorter one.
In standard form, \( b^2 > a^2 \) signifies that the major axis is vertical, so the calculations align as follows:

• Given \( a^2 = 3 \), we deduce \( a = \sqrt{3} \).
• Given \( b^2 = 9 \), we deduce \( b = 3 \).
• The length of the major axis is twice \( b \), so \( 2b = 6 \).
• The length of the minor axis is twice \( a \), so \( 2a = 2\sqrt{3} \).

These lengths provide the full span of the ellipse in both vertical and horizontal directions. The axis lengths play a crucial role in sketching out the size and shape of the ellipse.

Foci of Ellipse

The foci of an ellipse are two particular points located symmetrically along the major axis. They have a special relationship to the shape of the ellipse, as the sum of distances from the foci to any point on the ellipse is constant.
To determine the foci, use the equation \( c^2 = b^2 - a^2 \). In this example:

• Calculate \( c^2 \) as \( 9 - 3 = 6 \).
• Thus, \( c = \sqrt{6} \).
• The coordinates for the foci are \((0, \sqrt{6})\) and \((0, -\sqrt{6})\).

The foci help in defining the elliptical shape and are pivotal for understanding how ellipses behave in conic sections.

Standard Form

Ellipses are often represented in a standard form because it simplifies the process of identifying key properties, such as major and minor axes, center, and foci.
The general standard form is\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \].
For the equation \( 27x^2 + 9y^2 = 81 \), rewriting in standard form means dividing each term by 81, yielding \[ \frac{x^2}{3} + \frac{y^2}{9} = 1 \].
This form makes it straightforward to identify:

• The center of the ellipse at \((0, 0)\)
• \( a^2 = 3 \) and \( b^2 = 9 \), indicating a vertical major axis

The clarity provided by the standard form allows for easy determination of other characteristics and makes subsequent calculations more manageable.

Conic Sections

Ellipses are part of a broader category of curves known as conic sections, formed by the intersection of a plane and a cone.
This family of curves also includes circles, parabolas, and hyperbolas, each arising under different conditions of the intersecting plane. An ellipse is formed when the plane cuts through both nappes of the cone at a shallow angle relative to the base.
Key attributes include:

• The total distance from one point on the ellipse to both foci is constant.
• Ellipses have a distinct flattening along their major and minor axes.

The study of conic sections, including ellipses, plays a significant role in a variety of fields, from physics to engineering, because they describe the paths of objects under simple gravitational forces.