Problem 9 - Precalculus

Question

Graph each conic section. If the conic is a parabola, specify (using rectangular coordinates) the vertex and the directrix. If the conic is an ellipse, specify the center, the eccentricity, and the lengths of the major and minor axes. If the conic is a hyperbola, specify the center, the eccentricity, and the lengths of the transverse and conjugate axes. $$r=\frac{8}{5+3 \sin \theta}$$

Step-by-Step Solution

Answer

It's an ellipse with center (0,0), eccentricity 0.6, major axis 5, and minor axis 4.

Step 1: Determine the Type of Conic Section

The given equation is in the form of \( r = \frac{ed}{1 + e \sin \theta} \), which suggests it is a conic with the directrix perpendicular to the polar axis. The general form is \( r = \frac{ed}{1 + e \sin \theta} \) or \( r = \frac{ed}{1 - e \cos \theta} \) for conic sections in polar coordinates. Since the formula matches \( r = \frac{ed}{1 + e \sin \theta} \), it indicates the conic's directrix is horizontal. We recognize this specific form as an equation of a conic represented in polar coordinates.

Step 2: Identify the Conic Type Using Eccentricity

In the equation \( r = \frac{8}{5 + 3 \sin \theta} \), identify \( e = \frac{3}{5} \) from the coefficient of \( \sin \theta \). The value of \( e \) (eccentricity) determines the conic type:- If \( e = 1 \), it's a parabola.- If \( e < 1 \), it's an ellipse.- If \( e > 1 \), it's a hyperbola.Here, \( e = 0.6 \), which is less than one, indicating the conic section is an ellipse.

Step 3: Determine Characteristics of the Ellipse

Now that we've established the conic is an ellipse, we'll find its characteristics:- **Center**: The center of the ellipse is at the pole \((0, 0)\) because the equation is in the form \( r = \frac{ed}{1 + e \sin \theta} \).- **Eccentricity (e)**: Already given as \(0.6\).- **Major and minor axes**: The length of the major axis is given by \( \frac{2d}{1 - e^2} \), and the minor axis is given by \( \frac{2d}{\sqrt{1 - e^2}} \). The parameter \( d = \frac{8}{5} = 1.6 \) (from the polarization constant, since \( r(\theta = \frac{\pi}{2}) = \frac{8}{3} \)) gives us: - Length of the major axis: \( \frac{2 \times 1.6}{1 - (0.6)^2} = \frac{3.2}{0.64} = 5 \) - Length of the minor axis: \( \frac{3.2}{\sqrt{0.64}} = \frac{3.2}{0.8} = 4 \)

Key Concepts

Eccentricity in Ellipses

The concept of eccentricity is essential in understanding different types of conic sections. In the context of an ellipse, eccentricity (\( e \)) quantifies how much the shape of the ellipse deviates from that of a perfect circle. Eccentricity is a value between 0 and 1 for ellipses. When the eccentricity is close to 0, the ellipse looks more like a circle. As the eccentricity increases towards 1, the ellipse becomes more elongated.

For the given equation, \( r = \frac{8}{5 + 3 \sin \theta} \), the eccentricity \( e \) is calculated as \( \frac{3}{5} \), which is 0.6. Since this value is less than 1, it confirms that the conic section is indeed an ellipse.

Understanding eccentricity is crucial as it helps classify conic sections into different types:

• If \( e = 0 \), it denotes a circle.
• If \( e < 1 \), the shape is an ellipse.
• If \( e = 1 \), it represents a parabola.
• If \( e > 1 \), the conic is a hyperbola.

Ellipses in Polar Coordinates

Polar coordinates are often used to express conic sections, providing a different viewpoint compared to the more familiar Cartesian system. In polar coordinates, the position of a point is determined by an angle and a distance from a reference point, usually the origin. This system is especially useful for conic sections centered at the origin.

The equation of an ellipse in polar coordinates often takes the form \( r = \frac{ed}{1 + e \sin \theta} \) or \( r = \frac{ed}{1 + e \cos \theta} \). Here, \( d \) is a constant reflecting the size or scale of the ellipse, and \( e \) is the eccentricity. In our example, \( r = \frac{8}{5 + 3 \sin \theta} \), we identify \( d = \frac{8}{5} \) and use \( \sin \theta \), indicating the major axis is directed vertically because the \( \sin \) component typically describes vertical orientation.

Utilizing polar coordinates allows for straightforward calculations and provides insights into the geometric properties of the ellipse.

Major and Minor Axes

In ellipses, the terms major and minor axes refer to the longest and shortest diameters of the ellipse, respectively. These axes are crucial for understanding the precise shape and orientation of an ellipse.

The major axis is the longest diameter of the ellipse, stretching from one end to the opposite end through the center. In our problem, the length of the major axis is calculated using the formula \( \frac{2d}{1 - e^2} \), giving us a length of 5 units.

The minor axis is perpendicular to the major axis at the center and is the shortest diameter of the ellipse. In this case, the length of the minor axis is calculated as \( \frac{2d}{\sqrt{1 - e^2}} = 4 \) units.

Knowing the lengths of these axes allows you to plot the shape of the ellipse accurately. The center, being the point where these axes intersect, is located at the origin in polar coordinates in this scenario, simplifying calculations and visualizations.