Problem 32 - Calculus Early Transcendentals

Question

Sketch a 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. $$12 x^{2}+5 y^{2}=60$$

Step-by-Step Solution

Answer

Question: Sketch the graph of the given ellipse \(12x^2 + 5y^2 = 60\) and identify the coordinates of the vertices and foci, as well as the lengths of the major and minor axes. Answer: The standard form of the ellipse is \(\frac{x^2}{5} + \frac{y^2}{12} = 1\). The lengths of the major and minor axes are \(2\sqrt{12}\) and \(2\sqrt{5}\), respectively. The coordinates of the vertices are \((0, \sqrt{12})\) and \((0, -\sqrt{12})\), while the coordinates of the foci are \((0, \sqrt{7})\) and \((0, -\sqrt{7})\).

Step 1: Rewrite the equation in the standard form

Divide both sides of the equation by 60 to obtain the standard form: $$\frac{x^2}{5} + \frac{y^2}{12} = 1$$

Step 2: Identify the major and minor axes' lengths

Since the larger denominator is under \(y^2\), the major axis is vertical. Therefore, we will compare the ellipse equation with the standard form of a vertical ellipse: $$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$$ Here, \(a^2 = 12\) and \(b^2 = 5\). Thus, we have \(a = \sqrt{12}\) and \(b =\sqrt{5}\).

Step 3: Calculate the distance between the foci

To find the distance between the foci, we use the formula \(c = \sqrt{a^2 - b^2}\). In this case: $$c = \sqrt{12-5} = \sqrt{7}$$

Step 4: Identify the coordinates of the vertices and foci

Vertices are found at \((0, a)\) and \((0, -a)\). In this case, they are \((0, \sqrt{12})\) and \((0, -\sqrt{12})\). Since it is a vertical ellipse, the foci are found at \((0, c)\) and \((0, -c)\). In this case, they are \((0, \sqrt{7})\) and \((0, -\sqrt{7})\).

Step 5: Sketch the graph

Plot the center of the ellipse at \((0,0)\). Next, plot the vertices and the foci on the graph using their coordinates. Now draw the ellipse by joining and smoothing the points, keeping in mind that the distance between the foci remains constant along the major axis. Lastly, check your work using a graphing utility.

Key Concepts

Vertices of an Ellipse

The vertices of an ellipse are key points where the shape changes direction along its major axis. For a vertical ellipse, like in the given example, the vertices fall on the y-axis. They are calculated using the values of 'a' and 'b'. These terms represent semi-major and semi-minor axes, respectively.

To find the coordinates of the vertices, especially for our vertical ellipse \(\frac{x^2}{5} + \frac{y^2}{12} = 1\), we identify it has vertices at \(0, a\) and \(0, -a\).
Here, \(a = \sqrt{12}\). Thus, the vertices are at the points \(0, \sqrt{12}\) and \(0, -\sqrt{12}\).

Key steps:

• Determine 'a' as the square root of the larger denominator (12) in your ellipse equation.
• Position vertices along the vertical (y) axis, showing the maximum width of the ellipse.

This identification of vertices is crucial for graphing the ellipse accurately.

Foci of an Ellipse

While vertices mark the broadest parts of an ellipse, the foci are the internal compass points used to define the shape, especially relevant in practical applications like satellite orbits.

For ellipses, the relationship between distances from any point on the ellipse to the two foci is constant. To calculate the foci's position for our ellipse, we use the formula \(c = \sqrt{a^2 - b^2}\), where 'a' and 'b' are derived from the ellipse equation.

In our example with \(\frac{x^2}{5} + \frac{y^2}{12} = 1\)

• 'a' value originates from \(a^2 = 12\) so \(a = \sqrt{12}\)
• 'b' value originates from \(b^2 = 5\) so \(b = \sqrt{5}\)
• 'c' is determined using the foci formula, resulting in \(c = \sqrt{7}\).

Therefore, for our scenario, the foci are at \(0, \sqrt{7}\) and \(0, -\sqrt{7}\), placed symmetrically along the y-axis, parallel to the major axis.

Graphing Ellipses

Drawing an ellipse involves more than just connecting dots. It's about representing the smooth, continuous nature of its curve.

The process starts with placing the center at origin \(0,0\) based on our simplified equation \(\frac{x^2}{5} + \frac{y^2}{12} = 1\). Here is a simple approach to graphing:

• Plot the Center: Begin by marking \(0,0\) on the graph.
• Identify Vertices: Position them at calculated values, such as \(0, \sqrt{12}\) above and \(0,-\sqrt{12}\) below the center.
• Determine and Mark Foci: These guide internal dimensions at \(0, \sqrt{7}\) and \(0, -\sqrt{7}\).
• Outline the Ellipse: Form the oval by smoothly connecting vertices, noting how the curve keeps equidistant from foci points.

Using graphing utilities can reaffirm your sketch. By inputting the ellipse's equation, you can ensure correctness through digital verification. Understanding these parts contributes to competence in managing complex mathematical concepts like ellipses.