Problem 22 - Precalculus: Mathematical for Calculus

Question

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{8}{3+\cos \theta}$$

Step-by-Step Solution

Answer

The conic is a parabola with eccentricity 1, vertex (4,0), and focus (0,0).

Step 1: Identify the Conic Section Formula

The given polar equation is \( r = \frac{8}{3 + \cos \theta} \). The formula for a conic in polar coordinates is \( r = \frac{ed}{1 + e \cos \theta} \) or \( r = \frac{ed}{1 + e \sin \theta} \). Comparing, we have \( e = 1 \) and \( ed = 8 \).

Step 2: Calculate Eccentricity

From the comparison in Step 1, we have \( 3 + e \cos \theta = 3 + \cos \theta \). So, \( e = 1 \). This implies the conic is a parabola because its eccentricity \( e = 1 \).

Step 3: Find Directrix and Focus

Given \( ed = 8 \) and \( e = 1 \), we find \( d = 8 \). For a parabola in polar coordinates \( r = \frac{ed}{1 + e \cos \theta} \), the vertex is at \( r = \frac{d}{2} = \frac{8}{2} = 4 \) along the positive x-axis, and the directrix is the line \( x = -4 \). The vertex is at (4,0) and focus at (0,0).

Step 4: Sketch the Conic

Sketch a parabola opening to the left. Label the vertex at \((4, 0)\), and the directrix as a vertical line at \( x = -4 \). Ensure the parabola is centered at the origin with the opening aligned as stated.

Key Concepts

Eccentricity

Eccentricity is a key parameter that helps classify conic sections such as circles, parabolas, ellipses, and hyperbolas. In simple terms, eccentricity (\(e\)) measures how much a conic section deviates from being circular.

• For a circle, the eccentricity is 0, which means it's perfectly round.
• For an ellipse, the eccentricity is between 0 and 1.
• A parabola has an eccentricity of 1.
• Hyperbolas have an eccentricity greater than 1.

In the context of the given exercise, the given polar equation has an eccentricity of \(e = 1\), which uniquely identifies it as a parabola. This understanding that \(e = 1\) directly signals a parabolic shape is crucial when analyzing polar equations.

Parabola

A parabola is a unique conic section characterized by its symmetrical and U-shaped appearance. It is defined by a specific focus and directrix. The focus of a parabola is a fixed point, while the directrix is a fixed line. The parabola consists of all points equidistant to both the focus and directrix.In the polar coordinate system, a parabola with an eccentricity of 1 can be expressed with equations like \(r = \frac{ed}{1 + e \cos \theta}\) or \(r = \frac{ed}{1 + e \sin \theta}\). This shape can be neatly described by the equation given in the exercise \(r = \frac{8}{3 + \cos \theta}\).

• Here, \(e = 1\) confirms it's a parabola.
• The vertex, calculated from the given formula, is located at \((4, 0)\) in polar coordinates.
• The directrix is found as the line \(x = -4\)

A parabola like this will open to the left when sketched on the Cartesian plane, which is visually distinctive.

Conic Section

Conic sections are fundamentally important shapes in mathematics that arise from slicing a cone with a plane. These shapes include circles, ellipses, parabolas, and hyperbolas.

• A circle is the most simple and symmetrical conic section.
• An ellipse looks like a stretched circle.
• A hyperbola consists of two mirror-image curves.
• A parabola, like the one in our context, is a section that forms a symmetric U-shape.

Conic sections can be understood using both Cartesian and polar coordinates, depending on the problem context. Polar coordinates define positions in a plane using a radius and angle, which is useful when dealing with rotational symmetries present in conics. The given problem requires interpreting the conic section within the polar coordinates framework, which provides insights into the overall geometry. Analyzing conics in this way is especially helpful for understanding their properties and how they relate to each other geometrically. Understanding how polar equations translate into familiar shapes is a vital aspect of mastering conic sections.