Problem 14 - Thomas Calculus

Question

Exercises \(13-16\) give foci and corresponding directrices of ellipses centered at the origin of the \(x y\) -plane. In each case, use the dimensions in Figure 10.19 to find the eccentricity of the ellipse. Then find the ellipse's standard-form equation. $$ \begin{array}{l}{\text { Focus: }(4,0)} \\ {\text { Directrix: } \quad x=\frac{16}{3}}\end{array} $$

Step-by-Step Solution

Answer

Eccentricity is \( \frac{\sqrt{3}}{2} \). The standard form equation is \( \frac{3x^2}{64} + \frac{3y^2}{16} = 1 \).

Step 1: Understanding Eccentricity

The eccentricity of an ellipse, denoted as \( e \), is a measure of how much the ellipse deviates from being a circle. The closer the eccentricity is to 0, the more circular the ellipse. The eccentricity is found using the relationship: \( e = \frac{c}{a} \), where \( c \) is the distance from the center to a focus and \( a \) is the distance from the center to a vertex on the major axis.

Step 2: Identify Known Values

We know from the problem that the focus is at \((4,0)\). This tells us that \( c = 4 \). The directrix is given as \( x = \frac{16}{3} \), which is the distance \( \frac{a^2}{c} \) from the center of the ellipse to the directrix, allowing us to set up the equation \( \frac{a^2}{c} = \frac{16}{3} \).

Step 3: Solving for \( a \) and \( e \)

Using the directrix equation:\( \frac{a^2}{4} = \frac{16}{3} \).\( a^2 = \frac{64}{3} \). \( a = \sqrt{\frac{64}{3}} = \frac{8}{\sqrt{3}} \). Then, \( e = \frac{c}{a} = \frac{4}{\frac{8}{\sqrt{3}}} = \frac{\sqrt{3}}{2} \). Thus, the eccentricity is \( e = \frac{\sqrt{3}}{2} \).

Step 4: Finding \(b\) to Define the Standard Equation

The relationship \( b^2 = a^2(1-e^2) \) helps to find \( b \). Hence, \( b^2 = \frac{64}{3}\left(1 - \left(\frac{\sqrt{3}}{2}\right)^2\right) = \frac{64}{3} \left(1 - \frac{3}{4}\right) = \frac{64}{3} \times \frac{1}{4} = \frac{16}{3} \). Thus, \( b = \sqrt{\frac{16}{3}} = \frac{4}{\sqrt{3}} \).

Step 5: Writing the Standard Form Equation

Now that we have \( a \), \( b \), and \( c \), the standard form of the equation of an ellipse centered at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Substituting the known values gives us \( \frac{x^2}{\frac{64}{3}} + \frac{y^2}{\frac{16}{3}} = 1 \), or after simplifying the fractions: \( \frac{3x^2}{64} + \frac{3y^2}{16} = 1 \).

Key Concepts

Standard Form Equation of an Ellipse

In order to understand the geometry of an ellipse, it is essential to grasp the concept of its standard form equation. An ellipse is characterized by its two axes: the major axis, the longest one, and the minor axis, the shortest. For ellipses centered at the origin, their general standard form equation is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Here, \( a \) represents the semi-major axis, and \( b \) represents the semi-minor axis. Both \( a \) and \( b \) are crucial in determining the elliptical shape, as they regulate its horizontal and vertical expansions respectively. The terms \( \frac{x^2}{a^2} \) and \( \frac{y^2}{b^2} \) explain how the ellipse stretches along the x-axis and y-axis.
To navigate the references:

• "\( x \)" and "\( a\)" relate to the major axis.
• "\( y \)" and "\( b\)" relate to the minor axis.

This straightforward equation describes not only the proportion the axes extend but also determines the symmetry about both x and y-axes.

Eccentricity Calculation

Eccentricity is one of the primary measures that define the shape of an ellipse. To define it, let's involve the symbol \( e \), which mathematically is calculated using the formula:
\[ e = \frac{c}{a} \]
Where \( c \) is the distance from the center to a focus, and \( a \) is the distance from the center to a vertex along the major axis.

• Eccentricity \( e \) ranges between 0 and 1.
• If \( e \) is close to 0, the ellipse appears almost circular.
• If \( e \) approaches 1, the ellipse is elongated.

Calculating \( a \) and \( c \) using the known values:- For a focus \((4,0)\), \( c = 4 \).- Given the directrix \( x = \frac{16}{3} \), using the relationship \( \frac{a^2}{c} = \frac{16}{3} \), we find \( a \).
The calculation unfolds with \( a = \frac{8}{\sqrt{3}} \), and inserting \( a \) and \( c \) in the formula provides:\[ e = \frac{\sqrt{3}}{2} \] Which signifies a moderately stretched ellipse not too elongated.

Ellipse Geometry Concepts

Understanding the geometry of an ellipse involves recognizing its symmetry and proportional dimensions.- The longest horizontal spread is specified as the major axis. Conversely, the shortest vertical stretch is the minor axis.- An ellipse has two foci located symmetrically on the major axis, crucial to its unique properties that distinguish it from other conic sections such as parabolas or hyperbolas.Geometrical Features:- **Vertices**: Points at the ends of the major axis.- **Co-vertices**: Ends of the minor axis.- **Foci**: Not fixed at the midpoint but placed more towards the vertices, governing the bending of the ellipse.The directrix contributes to its shape and position:- It assists in regulating the orientation of the ellipse relative to its center.- The distance formula \( \frac{a^2}{c} = \text{distance to the directrix} \) interconnects with the focal distance \( c \) assisting in eccentricity predictions.In essence, these elements work together to describe not just the size but also how rounded or stretched the ellipse is, and the unique symmetry that characterizes its form.