Question

For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. $$ r=\frac{-1}{1+\cos \theta} $$

Short answer

The eccentricity is 1, so the conic is a parabola.

Step-by-step solution

Identify the Form

The given polar equation is \( r = \frac{-1}{1 + \cos \theta} \). This equation follows the standard form of a conic in polar coordinates: \( r = \frac{ed}{1 + e\cos\theta} \), where \( e \) is the eccentricity and \( d \) is a constant. Notice that it closely resembles the standard form except the numerator is \(-1\).

Determine the Eccentricity

In the given equation, compare it with the standard form \( r = \frac{ed}{1 + e\cos\theta} \). Here, \( ed = -1 \) and the denominator \(1 + e\cos\theta\) suggests \( e \) is the coefficient of \( \cos\theta \). In the equation, the terms imply \( e = 1 \). Thus, the eccentricity \( e = 1 \).

Identify the Conic Type

The value of the eccentricity \( e \) determines the type of conic: \( e = 0 \) is a circle, \( 0 < e < 1 \) is an ellipse, \( e = 1 \) is a parabola, and \( e > 1 \) is a hyperbola. Since the eccentricity \( e \) for this equation is 1, the conic is a parabola.

Key concepts

Eccentricity

Eccentricity is a fundamental concept in conic sections, defining the shape of the conic. It's represented by the symbol \( e \) and determines whether a conic is a circle, an ellipse, a parabola, or a hyperbola. Each of these shapes has a unique range of eccentricity values:

• For a circle, the eccentricity is \( e = 0 \).
• An ellipse has eccentricity values such that \( 0 < e < 1 \).
• A parabola has eccentricity exactly \( e = 1 \).
• A hyperbola has eccentricity greater than one, \( e > 1 \).

In the context of the given problem, the equation \( r = \frac{-1}{1 + \cos \theta} \) follows a polar form that lets us equate and determine the eccentricity. By matching terms, we find that the eccentricity \( e \) is 1, indicating that the conic is a parabola. Understanding these concepts helps distinguish between conic types seamlessly.

Conics Identification

Identifying the type of conic based on its equation is crucial in solving and understanding polar conics. In mathematics, conics like circles, ellipses, parabolas, and hyperbolas can all be defined through parameters, of which eccentricity is vital.

To identify a conic section using polar equations, focus on the form \( r = \frac{ed}{1 + e \cos \theta} \) or \( r = \frac{ed}{1 + e \sin \theta} \). Check the numerator and the coefficients. In our case, with \( e = 1 \), we instantly know that the conic is a parabola. Knowing this, students can apply the same principles to assess and classify various polar equations effectively.

Polar Equations

Polar equations allow for a different, yet insightful, perspective on conics. Unlike Cartesian coordinates, polar coordinates express the location of a point by means of an angle and a radius from a central origin point. This form is especially accommodating for curves that surround a center point, such as conics.

• The standard polar form \( r = \frac{ed}{1 + e \cos \theta} \) or \( r = \frac{ed}{1 + e \sin \theta} \) generally simplifies analysis of conics.
• Denominators with \( \cos \theta \) or \( \sin \theta \) help in determining orientation—horizontal or vertical —of the conic.
• Negative numerators suggest reflection or specific axis orientations, but don't alter the conic type.

The given exercise illustrates a profound aspect of polar conic equations—how changing variables and components relate directly to the geometric properties they describe, like the eccentricity and type of conic.