Problem 17 - Precalculus: Mathematical for Calculus

Question

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{2}{1-\cos \theta}$$

Step-by-Step Solution

Answer

The eccentricity is 1, thus it's a parabola.

Step 1: Identify the Conic Type

The given equation is in the form \[ r = \frac{ed}{1 + e \cos \theta} \]which represents a conic section, where "e" is the eccentricity and "d" is a constant. Comparing this with our equation \[ r = \frac{2}{1 - \cos \theta} \]we can see that the formula has \( e = 1 \) and \( d = 2 \). Thus, this is a conic section with a focus at the origin.

Step 2: Find the Eccentricity

The eccentricity \( e \) identifies the type of conic. In our equation: \( r = \frac{2}{1 - \cos \theta} \), the eccentricity is: \( e = 1 \). For ellipses, \( 0 < e < 1 \); for parabolas, \( e = 1 \); and for hyperbolas, \( e > 1 \). Therefore, this conic is a parabola since \( e = 1 \).

Step 3: Confirm the Conic Type

With the eccentricity value \( e = 1 \), the conic is classified as a parabola. A parabola has the characteristic that its eccentricity is equal to 1.

Step 4: Sketch the Parabola

To sketch the parabola given by: \( r = \frac{2}{1 - \cos \theta} \):1. Recognize that this is a polar equation for a conic section with its focus at the pole (origin). 2. Recall that a parabola with this form opens in the direction of decreasing \( \theta \), starting from \( \theta = 0 \).3. The vertex of this parabola is at the point where \( \theta = 0 \), which corresponds to \( r = 2/ (1 - 1) \rightarrow \mathrm{infinity} \). Thus, adjust \( \theta \) slightly to see the behavior of the graph. But generally, parabolas open outward.

Step 5: Label the Vertex

The vertex of the parabola in polar coordinates is at the position where \( \theta \to 0^{+} \). However, since the infinite point at \( \theta = 0 \) doesn't contribute clearly for sketching in traditional terms, we place the vertex at a position on the horizontal axis approaching this line as \( r \) increases rapidly (an approximation as the graph becomes clearer graphically).

Key Concepts

Conic Section

Conic sections are fascinating shapes that come from cutting a cone at different angles. These sections include circles, ellipses, parabolas, and hyperbolas.
Each conic has unique properties depending on its eccentricity value.

• If the eccentricity (\( e \)) is 0, the conic is a circle.
• If \( 0 < e < 1 \), it's an ellipse.
• If \( e = 1 \), it forms a parabola, like in our exercise.
• If \( e > 1 \), it creates a hyperbola.

Understanding these properties helps us identify which conic type we are dealing with by just examining the eccentricity.

Polar Coordinates

Polar coordinates offer an alternative way to locate points in a two-dimensional space. Unlike the familiar Cartesian coordinates, which use \( x \) and \( y \) axes, polar coordinates utilize a radius and an angle.

• The radius (\( r \)) is the distance of the point from the origin.
• The angle (\( \theta \)) is measured from the positive \( x \)-axis.

Equations in polar coordinates often take forms like \( r = \frac{a}{1 + b \cos \theta} \), which is essential for describing conic sections centered around the origin, especially when the focus is on the pole.

Parabola

A parabola is a unique type of conic section defined by its characteristic that the eccentricity (\( e \)) equals 1.

• In our exercise, the given equation \( r = \frac{2}{1 - \cos \theta} \) signifies a parabola.
• A defining feature is its shape, which is symmetric and curved.

The parabolic curve in polar coordinates has its focus at the origin, opening outward. Parabolas are special because, unlike ellipses or hyperbolas, they continue indefinitely in one direction. Understanding parabolas' symmetry and openness helps us sketch them accurately.

Focus

The focus is a critical point that helps define the shape of certain conic sections. In polar coordinates, especially for a parabola, the focus is at the origin, also known as the pole.

• This focus determines how the conic section behaves and where it is positioned.
• In our parabola, described by the equation \( r = \frac{2}{1 - \cos \theta} \), the focus is crucial for orienting the parabola's opening.
• The focus is the point from which the parabola "spreads out." It acts like a guiding light for the conic's growth along its path.

Understanding the role of the focus is essential in visualizing and working with conic sections, especially when graphing them in polar coordinates.