Problem 81 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. You told me that an ellipse centered at the origin has vertices at \((-5,0)\) and \((5,0),\) so I was able to graph the ellipse.

Step-by-Step Solution

Answer

The statement makes sense. The ellipse can be graphed with the given information (origin and vertices). However, without the 'b' value, the exact shape of the ellipse cannot be accurately represented.

Step 1: Understanding the Properties of an Ellipse

An ellipse centered at the origin can be represented by the equation \((\frac{x^2}{a^2}) + (\frac{y^2}{b^2}) = 1\), where 'a' is the semi-major axis and 'b' is the semi-minor axis. If the vertices are given as (-5,0) and (5,0), it signifies that the ellipse is stretched more along the x-axis.

Step 2: Determining ‘a’ and ‘b’ using Vertices

The vertices of the ellipse are at a distance 'a' along the x-axis from the center of the ellipse. Here, the vertices (-5, 0) and (5, 0) indicate that 'a' is 5. Since only the vertices (major axis endpoints) are given and not the co-vertices (minor axis endpoints), 'b' cannot be determined from the given information.

Step 3: Evaluating the Feasibility of the Statement

From the given statement, the individual was able to graph the ellipse knowing that it's centered at the origin with 'a' equals to 5. Although 'b' isn't given, the ellipse can still be sketched on a graph. One just has to remember that the exact shape of the ellipse (whether it's more elongated or more circular) can't be determined without the 'b' value.

Key Concepts

Semi-Major Axis

In an ellipse, the semi-major axis is the longest radius and defines the maximum extent of the ellipse. It represents half of the longest diameter across the ellipse. For an ellipse centered at the origin, this axis can lie either horizontally or vertically, depending on the layout of the vertices.

• The equation of the ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a \) is the semi-major axis.
• In the given example, vertices at \((-5,0)\) and \( (5,0) \) indicate that the semi-major axis is along the x-axis, and \( a = 5 \).

This means that the ellipse stretches out to 5 units along either side of the x-axis from the center.

Semi-Minor Axis

The semi-minor axis is the shortest radius of the ellipse, perpendicular to the semi-major axis. It is half of the shortest diameter of the ellipse. In many cases, the value of the semi-minor axis 'b' is not directly given and must be calculated or inferred from additional information.

• It appears in the equation of an ellipse as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) alongside the semi-major axis.
• In the exercise, \( b \) is not provided. Without additional points indicating width along the y-axis, 'b' remains undetermined.

Having 'b' helps define the exact shape, telling us whether the ellipse is more circular or elongated.

Vertices

Vertices are key points on an ellipse that define its overall size and orientation. They are located at the endpoints of the major axis. For an ellipse centered at the origin, they lie symmetrically around the center.

• Vertices are the points \( (a, 0) \) and \((-a, 0)\) or \( (0, a) \) and \( (0, -a) \) for horizontal and vertical ellipses, respectively.
• Given vertices \((-5,0)\) and \((5,0)\) imply the ellipse is horizontal.

These points are crucial in graphing the ellipse and help in identifying the semi-major axis length.

Graphing Ellipses

Graphing ellipses involves plotting points based on the properties of the ellipse's equation. Starting with the center, the vertices and co-vertices guide the shape. The graph will take a unique appearance based on the lengths of 'a' and 'b'.

• To graph, identify and plot the center, then use the values of \( a \) and \( b \) to find vertices and co-vertices.
• The given ellipse has a known \( a \) and unknown \( b \), but a rough graph can still be created using the known vertices.

While more information provides precision, understanding how to use given points allows students to sketch reasonable approximations.