Problem 41 - Algebra 2

Question

The eccentricity of an ellipse is a measure of how nearly circular it is. Eccentricity is defined as \(\frac{s}{a},\) where \(c\) is the distance from the center to a focus and \(a\) is the distance from the center to a vertex. a. Find the eccentricity of an ellipse with foci \(( \pm 9,0)\) and vertices \(( \pm 10,0) .\) Sketch the graph. b. Find the eccentricity of an ellipse with foci \(( \pm 1,0)\) and vertices \(( \pm 10,0)\) Sketch the graph. c. Describe the shape of an ellipse that has an eccentricity close to \(0 .\) d. Describe the shape of an ellipse that has an eccentricity close to \(1 .\)

Step-by-Step Solution

Answer

a. The eccentricity of an ellipse with foci at \(\pm 9\) and vertices at \(\pm 10\) is \(0.9\). It's more elongated. b. The eccentricity of an ellipse with foci at \(\pm 1\) and vertices at \(\pm 10\) is \(0.1\). It's almost circular. c. An ellipse with eccentricity close to 0 is almost a circle. d. An ellipse with eccentricity close to 1 is more elongated.

Step 1: Calculate Eccentricity - Ellipse 1

Use the given values for foci \(\pm 9\) (which is \(c\)) and vertices \(\pm 10\) (which is \(a\)) in the formula for eccentricity. Therefore, eccentricity \(e = c/a = 9/10 = 0.9\).

Step 2: Calculate Eccentricity - Ellipse 2

Use the given values for foci \(\pm 1\) (which is \(c\)) and vertices \(\pm 10\) (which is \(a\)) in the formula for eccentricity. Therefore, eccentricity \(e = c/a = 1/10 = 0.1\).

Step 3: Describe Shape - Eccentricity Close to 0

For an ellipse with eccentricity close to 0, the shape will be almost a circle. This is because the foci are very close together, meaning the two halves of the ellipse are almost the same size.

Step 4: Describe Shape - Eccentricity Close to 1

For an ellipse with eccentricity close to 1, the ellipse will be elongated. The ellipse gets closer to becoming a line as the foci move further apart. This gives the two halves more unequal sizes and the ellipse becomes longer.

Key Concepts

Conic Sections

Conic sections are shapes that are created as a plane intersects a cone. They are fundamental concepts in geometry and appear in many areas of mathematics. There are four primary types of conic sections:

• Circle
• Ellipse
• Parabola
• Hyperbola

Each conic section has unique properties depending on the angle at which the plane cuts through the cone. Ellipses, for example, occur when the plane cuts through the cone at an angle, but not steep enough to intersect both halves. The result is an oval shape, which can vary from nearly circular to very elongated depending on its eccentricity. Understanding conic sections is crucial as they represent various physical and theoretical phenomena in nature.

Ellipse Properties

An ellipse is a fascinating geometric figure defined by certain unique properties. It is oval-shaped and is characterized by two fixed points known as foci.

• The sum of the distances from any point on the ellipse to the two foci is constant.
• The longest diameter across the ellipse is called the major axis, and the shortest is the minor axis.
• The vertices are the endpoints of the major axis.

A distinctive property of an ellipse is its symmetry along both its major and minor axes. The distance from the center of the ellipse to one of its foci is denoted as \(c\), and the distance to a vertex is represented by \(a\). These distances are crucial in calculating the ellipse's eccentricity, which indicates how stretched or circular the ellipse is.

Eccentricity Calculation

Eccentricity is a measure of how much an ellipse deviates from being a circle. It is computed using the formula:\[ e = \frac{c}{a} \]where \(c\) is the distance from the center to a focus and \(a\) is the distance from the center to a vertex of the ellipse. Eccentricity values range between 0 and 1.

• An eccentricity of 0 means the ellipse is a perfect circle.
• A low eccentricity, such as 0.1, results in an almost circular shape.
• A high eccentricity, nearing 1, indicates a more elongated shape, almost line-like.

To find the eccentricity, simply divide the distance to the focus by the distance to the vertex. In our examples, the first ellipse has an eccentricity of 0.9, suggesting it is quite elongated, while the second ellipse, with an eccentricity of 0.1, is almost circular. Understanding eccentricity helps in visualizing how an ellipse appears and behaves geometrically.