Question

For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. $$ r=\frac{8}{2-\sin \theta} $$

Short answer

The eccentricity is 1/2, identifying the conic as an ellipse.

Step-by-step solution

Identify the Equation Form

The given polar equation is \( r = \frac{8}{2 - \sin \theta} \). This is in the form of a conic section equation in polar coordinates, which is \( r = \frac{ed}{1 + e\sin \theta} \) or \( r = \frac{ed}{1 - e\sin \theta} \).

Rewrite Equation to Standard Form

To match the standard conic form, rewrite the given equation \( r = \frac{8}{2 - \sin \theta} \) in the form \( r = \frac{ed}{1 - e\sin \theta} \), identifying the corresponding values. Here, we match the numerator and the part of the denominator.

Determine Eccentricity (e)

The equation \( r = \frac{8}{2 - \sin \theta} \) can be rewritten as \( r = \frac{8/2}{1 - (1/2)\sin \theta} \). Comparing it to \( r = \frac{ed}{1 - e\sin \theta} \), we see that \( ed = 8 \) and \( e\sin \theta = \sin \theta/2 \), which leads to \( e = 1/2 \).

Identify the Conic

The eccentricity \( e = 1/2 \) corresponds to an ellipse since for ellipses \( 0 < e < 1 \).

Key concepts

Eccentricity

Eccentricity is a fundamental parameter in the study of conic sections. In simple terms, it measures how much a conic section deviates from being circular. - An eccentricity of 0 indicates a perfect circle.- Values between 0 and 1 correspond to an ellipse.- An eccentricity of exactly 1 represents a parabola.- Eccentricities greater than 1 define a hyperbola. For example, in the polar equation given, we determined that the eccentricity, denoted by \( e \), is \( \frac{1}{2} \). This value shows it is part of an ellipse, as it falls between 0 and 1. Eccentricity plays a crucial role in classifying the type and properties of the conic section being analyzed. Calculating the eccentricity is often the first step in identifying these shapes.

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas. - **Circle**: Achieved when the plane's angle is perpendicular to the axis of the cone. - **Ellipse**: Formed when the plane cuts through the cone at an angle, creating an oval shape. - **Parabola**: Occurs when the plane is parallel to the slope of the cone. - **Hyperbola**: Created when the plane intersects both halves of the cone, resulting in two curves. These curves exist all around us in nature and engineering. The key to distinguishing them is through their eccentricity values and their geometric properties. In the context of polar equations, identifying the type of conic section comes from both its algebraic form and its calculated eccentricity.

Polar Coordinates

Polar coordinates offer a different way to locate a point in a plane using angles and radii, different from the typical Cartesian system of \( x \) and \( y \) coordinates. Instead of measuring how far and in which direction along the axes a point is, polar coordinates focus on:- The radius \( r \): The distance from the origin to the point.- The angle \( \theta \): Measured from the positive x-axis to the line connecting the point to the origin. Utilizing polar coordinates is particularly advantageous in situations involving circles and spirals, or when dealing with problems related to rotations around a central point. Polar coordinates allow for elegant expressions of conic sections, simplifying the analysis of curves like circles and ellipses. The polar equation \( r = \frac{8}{2 - \sin \theta} \) reflects such a setup, making it easier to interpret and solve through these lenses.

Ellipse

An ellipse is a type of conic section characterized by its oval shape and consists of all points where the sum of distances to two foci is constant. It is the shape observed when a conic section has an eccentricity \( e \) that satisfies \( 0 < e < 1 \). Some elliptical properties include:- **Major and Minor Axes**: The longest and shortest diameters of the ellipse.- **Foci**: Two central points that define the ellipse, different from the circle's singular center. Ellipses are commonly found in planetary orbits, where the gravitational focus lies at one focal point, with the other part almost empty. In determining the nature of our given polar equation and identifying it as an ellipse, the calculated eccentricity of \( \frac{1}{2} \) guided our understanding. Its polar representation helps in visualizing these conic sections by emphasizing the radii and angles that trace its perimeter.