Question

Sketch the graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$\frac{x^{2}}{5}+\frac{y^{2}}{7}=1$$

Short answer

Question: Sketch the graph of the ellipse given by the equation $$\frac{x^{2}}{5}+\frac{y^{2}}{7}=1$$ and identify its major and minor axes lengths, vertices, and foci. Answer: The graph of the ellipse is elongated along the y-axis with the major axis length being \(2\sqrt{7}\) and the minor axis length being \(2\sqrt{5}\). The vertices are \((\sqrt{5}, 0)\), \((-\sqrt{5}, 0)\), \((0, \sqrt{7})\), and \((0, -\sqrt{7})\), and the foci are \((0, \sqrt{2})\) and \((0, -\sqrt{2})\).

Step-by-step solution

1. Identifying the major and minor axes lengths

Given the equation of the ellipse: $$\frac{x^{2}}{5}+\frac{y^{2}}{7}=1$$ Notice that the denominator under \(y^2\) is larger than the denominator under \(x^2\), this tells us that the major axis is along the y-axis and the minor axis along the x-axis. The lengths of the major and minor axes are determined by twice the square root of the respective denominators: Minor axis length: \(2\sqrt{5}\) Major axis length: \(2\sqrt{7}\)

2. Identifying the vertices

To identify the vertices of the ellipse, we need to look at the points where the ellipse intersects its major and minor axes. Since the y-axis is the major axis, the vertices along the vertical direction will have x = 0, and the locations where the ellipse intersects the y-axis will be \(\pm \sqrt{7}\). Therefore, the vertices along the major axis are: \((0, \sqrt{7})\) and \((0, -\sqrt{7})\) For the minor axis vertices, the ellipse will intersect the x-axis at \(\pm \sqrt{5}\), with y = 0. Therefore, the vertices along the minor axis are: \((\sqrt{5}, 0)\) and \((-\sqrt{5}, 0)\)

3. Identifying the foci

To determine the locations of the foci, we will use the relationship: $$c = \sqrt{a^2 - b^2}$$ Where `c` is the distance from the ellipse's center to its foci, `a` is the distance from the center to the vertices along the major axis, and `b` is the distance from the center to the vertices along the minor axis. In this problem, \(a = \sqrt{7}\) and \(b = \sqrt{5}\), so we get: $$c = \sqrt{(\sqrt{7})^2 - (\sqrt{5})^2} = \sqrt{2}$$ Now, since the major axis is along the y-axis, the foci will be above and below the ellipse's center. Therefore, the foci are at \((0, \sqrt{2})\) and \((0, -\sqrt{2})\)

4. Sketch the graph of the ellipse

Following all the identified information: - Major and minor axes lengths: \(2\sqrt{7}\) and \(2\sqrt{5}\) respectively - Vertices: \((\sqrt{5}, 0)\), \((-\sqrt{5}, 0)\), \((0, \sqrt{7})\), and \((0, -\sqrt{7})\) - Foci: \((0, \sqrt{2})\) and \((0, -\sqrt{2})\) Now we can sketch the graph of the ellipse with this information. Use a graphing utility to check your work. The ellipse should be elongated along the y-axis, with the vertices and foci correctly labeled.

Key concepts

Vertices

Vertices are key points on an ellipse, where it intersects its major and minor axes. In simple terms, vertices are the farthest points from the center along these axes.
When working with an ellipse, such as the one given by \(\frac{x^2}{5}+\frac{y^2}{7}=1\), identifying vertices helps to understand its overall shape.

For this ellipse:

• The **major axis** is aligned with the y-axis. This is because the denominator of \(y^2\) (which is 7) is greater than that of \(x^2\) (which is 5).
• The vertices along the major axis are at points where \(x = 0\) and \(y = \pm \sqrt{7}\). Thus, their coordinates are \((0, \sqrt{7})\) and \((0, -\sqrt{7})\).
• On the **minor axis**, the ellipse crosses where \(y = 0\) and \(x = \pm \sqrt{5}\). Therefore, those vertices are \((\sqrt{5}, 0)\) and \((-\sqrt{5}, 0)\).

These vertices can act as reliable references when drawing or analyzing the ellipse.

Foci

The foci (singular: focus) of an ellipse are crucial in defining its shape and characteristics. These points lie along the major axis and are used in calculating the ellipse's eccentricity.
For our ellipsis \(\frac{x^2}{5}+\frac{y^2}{7}=1\), we calculate the foci using the formula:
\[c = \sqrt{a^2 - b^2}\] where \(a\) is \(\sqrt{7}\) (half the length of the major axis) and \(b\) is \(\sqrt{5}\) (half the length of the minor axis).

Solving for `c`:

• Substitute into the formula: \(c = \sqrt{\left(\sqrt{7}\right)^2 - \left(\sqrt{5}\right)^2} = \sqrt{2}\).

Because the major axis is on the y-axis, the foci are above and below the center at \(y = \pm \sqrt{2}\). Therefore, the coordinates of the foci are \((0, \sqrt{2})\) and \((0, -\sqrt{2})\).
This understanding of the ellipse's foci helps in visualizing the "flattening" or elongation of the shape along its major axis.

Major and Minor Axes

The major and minor axes are the backbone of an ellipse's structure. They define its longest and shortest dimensions.
The major axis is like the main "path," stretching from one end of the ellipse to another, passing through its center. Conversely, the minor axis is the shortest "path," also passing through the center but perpendicular to the major axis.

For our equation \(\frac{x^2}{5}+\frac{y^2}{7}=1\):

• The **length of the major axis** is computed as \(2\sqrt{7}\). This result derives from the larger denominator of \(y^2\), indicating it lies along the y-axis.
• Meanwhile, the **minor axis** length, \(2\sqrt{5}\), corresponds to \(x=\pm\sqrt{5}\), showing its alignment along the x-axis.

Knowing these lengths not only aids in graphing an accurate ellipse but also assists in spatial visualization of its dimensions, emphasizing an ellipse's elongated or flattened nature compared to a circle.