Problem 279 - Calculus Volume 3

Question

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

Step-by-Step Solution

Answer

The eccentricity is 1, indicating a parabola.

Step 1: Identifying the Standard Form

The given equation is in the form \( r = \frac{ed}{1 + e\cos\theta} \). Comparing it to standard polar equation \( r = \frac{ed}{1 + e\cos\theta} \), we see \( e = 1 \) and \( ed = -1 \).

Step 2: Identify Eccentricity

The eccentricity \( e \) is found to be 1 from the standard form equation comparison in Step 1.

Step 3: Classifying the Conic

The type of conic is determined by its eccentricity: \( e < 1 \) for ellipse, \( e = 1 \) for parabola, and \( e > 1 \) for hyperbola. Since \( e = 1 \), the conic is a parabola.

Key Concepts

Eccentricity

Eccentricity is a critical concept in the geometry of conics since it helps classify what type of conic section we are dealing with. It is denoted by the letter "\( e \)" and reflects how much the conic deviates from being circular.
For any conic section, the value of the eccentricity determines its shape:

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

In the context of the problem, when we identified that \( e = 1 \), this meant we were dealing with a parabola. Recognizing the eccentricity value helps students quickly identify and classify the conic section when analyzing its equations.

Polar Equations

Polar equations are another important concept when dealing with conics. In polar coordinates, each point on a plane can be determined by an angle and a distance from the origin. The standard form of a polar equation for a conic is:
\[ r = \frac{ed}{1 + e\cos\theta} \]
or
\[ r = \frac{ed}{1 + e\sin\theta} \]
In these equations, \( r \) is the radial distance from the origin, \( \theta \) is the angle, \( e \) represents eccentricity, and \( d \) is the directrix distance. By comparing the given conic's polar equation to the standard form, we determine specific parameters about the conic, like its eccentricity and orientation. This way, students can interpret and manipulate polar equations to find critical information about conic sections.

Classifying Conics

Classifying conics is crucial for understanding their properties and applications. Using the eccentricity \( e \) as a basis, conics are classified into four main types:

• Circles: These occur when \( e = 0 \). Circles are a special type of ellipse with no eccentricity.
• Ellipses: Occurring when \( 0 < e < 1 \), ellipses are elongated circles.
• Parabolas: When \( e = 1 \), the conic section is a parabola. In this case, the curve is open and both ends run parallel.
• Hyperbolas: These occur when \( e > 1 \). Hyperbolas have two open ends that run in opposite directions.

The classification helps determine how a conic will behave in various mathematical and physical contexts, and how it can be represented graphically. This understanding is key to solving more complex problems involving conic sections.