Problem 33 - Calculus Early Transcendentals

Question

Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. An ellipse whose major axis is on the \(x\) -axis with length 8 and whose minor axis has length 6

Step-by-Step Solution

Answer

Answer: The values of \(a\) and \(b\) are 4 and 3 respectively. The equation of the ellipse is \(\frac{x^2}{16} + \frac{y^2}{9} = 1\). The coordinates of the vertices are \((-4, 0)\) and \((4, 0)\), and the coordinates of the foci are \((-\sqrt{7}, 0)\) and \((\sqrt{7}, 0)\).

Step 1: Calculate \(a\) and \(b\) values

Half the length of the major axis is 8/2 = 4 and half the length of the minor axis is 6/2 = 3. Therefore, \(a=4\) and \(b=3\).

Step 2: Write the equation of the ellipse

Now we can plug \(a=4\) and \(b=3\) into the ellipse equation with center at the origin: \(\frac{x^2}{4^2} + \frac{y^2}{3^2} = 1\). So the equation of the ellipse is \(\frac{x^2}{16} + \frac{y^2}{9} = 1\).

Step 3: Find coordinates of the vertices

The vertices are located at \((\pm a, 0)\). Thus, the coordinates of the vertices are \((\pm 4, 0)\), or specifically, \((-4, 0)\) and \((4, 0)\).

Step 4: Calculate the distance between the center and foci

To find the distance \(c\) between the center and the foci, we use the relation \(c^2 = a^2 - b^2\). So, \(c^2 = 4^2 - 3^2 = 16 - 9 = 7\), and \(c = \sqrt{7}\).

Step 5: Find coordinates of the foci

The foci are located at \((\pm c, 0)\). Thus, the coordinates of the foci are \((\pm \sqrt{7}, 0)\), or specifically, \((-\sqrt{7}, 0)\) and \((\sqrt{7}, 0)\).

Step 6: Sketch the graph

To sketch the graph, plot the center at the origin (0,0), and the vertices at \((-4,0)\) and \((4,0)\). Mark the foci at \((-\sqrt{7},0)\) and \((\sqrt{7},0)\). Draw the ellipse, making sure that the major axis has a length of 8 and the minor axis has a length of 6. Label the vertices and foci on the graph.

Key Concepts

Vertices and Foci

The vertices and foci are essential parts of an ellipse, providing a clear understanding of its shape and orientation. Vertices are the points on the ellipse that lie farthest from the center along the major axis.
In our example, the major axis is along the x-axis, making the vertices

• at
  • (-4, 0)
  • (4, 0)

The foci of an ellipse are two fixed points inside the ellipse which are crucial for its definition. They provide balance around which the overall contour of the ellipse is established. In this case, the coordinates of the foci are:

• • (- \(\sqrt{7}\), 0)
  • (\(\sqrt{7}\), 0)

Understanding the positions of vertices and foci helps one grasp how stretched the ellipse is and in which direction it is oriented.

Major and Minor Axes

In an ellipse, the major axis is the longer diameter, while the minor axis is the shorter one. These two axes define the proportions and the elongation level of the ellipse.
The length of the major axis in this ellipse is 8. As it is aligned with the x-axis, the distance from the center to each end of major axis (the semi-major axis) is

• 4 units (because 8/2 = 4)

The length of the minor axis is 6, stretching along the y-axis. The semi-minor axis is

• 3 units (since 6/2 = 3) away from the origin.

This information tells us not only about the size of the ellipse but also frames our understanding regarding how the graph should look when drawn. The major and minor axes create a framework that the ellipse fits within.

Center at Origin

When an ellipse is centered at the origin, the coordinates of its center are at

• (0,0)

This creates a symmetry around both the x-axis and the y-axis.
For an ellipse aligned along the axes, the general equation with a center at the origin is given by:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where

• • \(a\) is the semi-major axis
  • \(b\) is the semi-minor axis

In our situation where the semi-major axis is 4 and semi-minor axis is 3, the equation of the ellipse becomes:\[\frac{x^2}{16} + \frac{y^2}{9} = 1\]Every part of the ellipse stems from this center point at the origin, creating a balanced, oval shape that can be easily graphed and analyzed.