Problem 12 - Thomas Calculus

Question

Exercises \(9-12\) give the foci or vertices and the eccentricities of ellipses centered at the origin of the \(x y\) -plane. In each case, find the ellipse's standard-form equation. $$ \begin{array}{l}{\text { Vertices: }( \pm 10,0)} \\ {\text { Eccentricity: } 0.24}\end{array} $$

Step-by-Step Solution

Answer

The ellipse's standard-form equation is \( \frac{x^2}{100} + \frac{y^2}{94.24} = 1 \).

Step 1: Understand Problem Keywords

Identify from the problem that we have been given the vertices and the eccentricity for an ellipse with its center at the origin. We need to find the standard equation of this ellipse.

Step 2: Recall the Standard Form of Ellipse Equation

The standard form of the equation of an ellipse centered at the origin is:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]for a horizontal major axis, where \(a\) is the semi-major axis (along the x-axis) and \(b\) is the semi-minor axis.

Step 3: Relate Given Vertices to Semi-major Axis

Since the vertices given are \(( \pm 10, 0)\), the distance from the center to a vertex along the x-axis is 10. Thus, \(a = 10\).

Step 4: Use Eccentricity to Find \(b\)

The eccentricity \(e\) of an ellipse is given by:\[ e = \frac{c}{a} \]where \(c\) is the distance from the center to a focus. We know \(e = 0.24\) and \(a = 10\), so:\[ 0.24 = \frac{c}{10} \]Solve for \(c\):\[ c = 2.4 \]

Step 5: Solve for \(b^2\) Using \(c^2 = a^2 - b^2\)

We use the relationship between \(a, b,\) and \(c\) in an ellipse:\[ c^2 = a^2 - b^2 \]Substitute the known values:\[ 2.4^2 = 10^2 - b^2 \]\[ 5.76 = 100 - b^2 \]\[ b^2 = 100 - 5.76 \]\[ b^2 = 94.24 \]

Step 6: Write the Standard Form Equation

Substitute \(a^2 = 100\) and \(b^2 = 94.24\) into the standard form equation:\[ \frac{x^2}{100} + \frac{y^2}{94.24} = 1 \]This is the ellipse's equation.

Key Concepts

Eccentricity

Eccentricity is a numerical value that describes how much an ellipse deviates from being a perfect circle. It is denoted as \(e\). This value ranges between 0 and 1. When the eccentricity is closer to 0, the shape of the ellipse is more like a circle. Conversely, when it approaches 1, the ellipse becomes elongated.

The formula for calculating the eccentricity of an ellipse is given by:

• \(e = \frac{c}{a}\)

where \(c\) is the distance from the center to a focus, and \(a\) is the semi-major axis length. In the given exercise, you can calculate \(c\) once you know \(e\) and \(a\). This relationship is crucial in finding the lengths of the axes and constructing the standard form of the ellipse equation.

Semi-major Axis

The semi-major axis of an ellipse is one of its two main radii and represents half the longest diameter or width. Typically, it is denoted by \(a\). In a horizontal ellipse that is centered at the origin, the semi-major axis extends along the x-axis.

• The standard form of an ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
• The value of \(a\) is found by measuring the distance from the center to any vertex.

In this problem, since the vertices are \((\pm 10, 0)\), the semi-major axis \(a\) is 10. Understanding this geometrical aspect is fundamental in correctly formulating the ellipse's equation.

Semi-minor Axis

The semi-minor axis is the other principal radius of an ellipse, perpendicular to the semi-major axis and shorter in length. It is denoted as \(b\). In an ellipse centered at the origin, it stretches along the y-axis in a horizontal orientation.To find \(b\), we can use the relationship between the semi-major axis \(a\), the semi-minor axis \(b\), and the distance \(c\) to a focus, as expressed by the equation:

• \(c^2 = a^2 - b^2\)
• From the given data: \(c = 2.4\), \(a = 10\).
• Calculate \(b^2\): \(b^2 = a^2 - c^2 = 100 - 5.76 = 94.24\).

Thus, the semi-minor axis gives us insight into the ellipse's shape and angle.

Standard Form of Ellipse

The standard form of the equation of an ellipse is a structured pattern used to describe its geometry. It simplifies expressing the relationship between the axes and the distances. For ellipses centered at the origin and aligned horizontally, the equation is given by:

• \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
• Where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

To derive this form, input the squared values of the axes:

• \(a^2 = 100\)
• \(b^2 = 94.24\)

After the calculation, the standard form of the ellipse is \(\frac{x^2}{100} + \frac{y^2}{94.24} = 1\). Knowing the standard form is essential as it is extensively used in mathematical computations involving ellipses.