Problem 61 - College Algebra with Modeling and Visualization

Question

Graph the ellipse. $$ \frac{x^{2}}{15}+\frac{y^{2}}{10}=1 $$

Step-by-Step Solution

Answer

The ellipse is centered at the origin, with vertices at \( (\pm \sqrt{15}, 0) \) and co-vertices at \( (0, \pm \sqrt{10}) \).

Step 1: Identify the equation form

The given ellipse equation is \( \frac{x^2}{15} + \frac{y^2}{10} = 1 \). This is in the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \( a^2 = 15 \) and \( b^2 = 10 \).

Step 2: Calculate the semi-major and semi-minor axes

The semi-major axis length \( a \) is given by \( a = \sqrt{15} \), and the semi-minor axis length \( b \) is given by \( b = \sqrt{10} \). Since \( a > b \), the major axis is along the x-axis.

Step 3: Determine the vertices

For an ellipse centered at the origin with the major axis along the x-axis, the vertices are \( (\pm a, 0) \). Therefore, the vertices are at \( (\pm \sqrt{15}, 0) \).

Step 4: Determine the co-vertices

The co-vertices are along the minor axis, which is the y-axis in this case. The co-vertices are \( (0, \pm b) \), or \( (0, \pm \sqrt{10}) \).

Step 5: Sketch the ellipse

Draw the x and y axes. Plot the vertices \( (\sqrt{15}, 0) \) and \( (-\sqrt{15}, 0) \), as well as the co-vertices \( (0, \sqrt{10}) \) and \( (0, -\sqrt{10}) \). Connect these points with a smooth, symmetrical oval shape to complete the ellipse.

Key Concepts

Standard Form of Ellipse

The standard form of an ellipse is an equation that identifies the geometric shape and its dimensions. For an ellipse centered at the origin, the equation typically looks like \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \( x \) and \( y \) are the variables representing the horizontal and vertical axes respectively. The terms \( a^2 \) and \( b^2 \) denote squares of the semi-major and semi-minor axes respectively.

An important distinction to remember is the orientation of the ellipse. If \( a^2 > b^2 \), the ellipse is stretched along the x-axis (horizontal ellipse), whereas if \( b^2 > a^2 \), it is stretched along the y-axis (vertical ellipse). This equation provides a clear, structured way to identify and graph ellipses easily.

• The equation must equal 1.
• Coefficients determine the axis lengths.
• The form signifies whether the ellipse is horizontal or vertical.

Semi-Major and Semi-Minor Axes

The semi-major and semi-minor axes are crucial components of an ellipse, providing its main dimensions. The semi-major axis is the longest radius of the ellipse, and the semi-minor axis is the shortest. To find these lengths in the standard equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), you calculate the square roots of \( a^2 \) and \( b^2 \).

Using the exercise's values:

• Semi-major axis: \( a = \sqrt{15} \). Since \( a > b \), the ellipse is horizontal.
• Semi-minor axis: \( b = \sqrt{10} \).

Knowing these axes helps in sketching the ellipse and understanding its shape better.

Vertices and Co-Vertices of Ellipse

Vertices and co-vertices are key locations on an ellipse that help define its shape and position. For an ellipse centered at the origin, its vertices are positioned along the major axis, while the co-vertices lie along the minor axis. In our exercise, with our horizontal ellipse, the vertices are \( (\pm \sqrt{15}, 0) \). These are the furthest points along the x-axis.

Co-vertices refer to the points that fall along the y-axis, where the minor axis is located. For this ellipse:

• Vertices: \( (\pm \sqrt{15}, 0) \)
• Co-vertices: \( (0, \pm \sqrt{10}) \)

These points are essential for sketching the ellipse accurately on a graph, providing reference points around which the oval shape is constructed.