Problem 28 - Precalculus: Mathematics for Calculus

Question

A polar equation of a conic is given. (a) Show that the conic is a hyperbola, and sketch its graph. (b) Find the vertices and directrix, and indicate them on the graph. (c) Find the center of the hyperbola, and sketch the $$r=\frac{6}{2+7 \cos \theta}$$

Step-by-Step Solution

Answer

The conic represented by the equation is a hyperbola; its vertices are located at distance \( \frac{12}{7} \) from the pole, and the directrix is at \( \frac{24}{49} \). The center is at the origin.

Step 1: Identify the Form of the Conic

The given polar equation is \( r = \frac{6}{2 + 7 \cos \theta} \). We notice it is in the form \( r = \frac{ed}{1 + e \cos \theta} \) where \( e \) is the eccentricity. Comparing, we can rewrite this as \( r = \frac{6/2}{1 + (7/2) \cos \theta} \) means \( e = \frac{7}{2} \). Since \( e > 1 \), the conic is a hyperbola.

Step 2: Find the Vertices and Directrix

In a polar form \(r = \frac{ed}{1 + e \cos \theta}\), the distance from the pole to the nearest vertex is \( d \), where \( ed = 6 \) implies \( d = \frac{6}{7/2} = \frac{12}{7} \). The directrix is parallel to the line of \( \theta = 0 \) and is located at a distance \( d/e = \frac{12}{7}/\frac{7}{2} = \frac{12}{7} \times \frac{2}{7} = \frac{24}{49} \) from the pole.

Step 3: Calculate the Center of the Hyperbola

For a hyperbola in polar form, the center is at the origin or pole, \( (0,0) \), because it's symmetric about the pole.

Step 4: Sketch the Graph

With \( e = \frac{7}{2} > 1 \), the hyperbola opens horizontally. Locate the vertex at \( (\frac{12}{7}, 0) \) and plot the directrix perpendicular to the initial line \( r = \frac{24}{49} \). The other branch symmetrically goes in the opposite direction because it's a hyperbola.

Key Concepts

Hyperbola

An understanding of hyperbolas is crucial for solving problems involving conic sections in polar coordinates. A hyperbola is a type of conic section that appears as an open curve, typically with two branches. In essence, it consists of two symmetrical parts that open in opposite directions. It's important to distinguish a hyperbola from an ellipse or parabola, which are other types of conics. The defining feature of a hyperbola is its eccentricity, which is greater than 1. This characteristic helps in classifying the conic section when analyzing the equation. In the context of polar coordinates, the general form \( r = \frac{ed}{1 + e \cos \theta} \) assists in identifying the conic as a hyperbola when \( e > 1 \).
In practical terms, hyperbolas are used in many areas of physics and engineering, such as determining satellite paths and in acoustics for sound wave behavior.

Eccentricity

Eccentricity is a measure of how much a conic section deviates from being circular. It's a crucial concept because it helps in identifying whether a conic is a circle, ellipse, parabola, or hyperbola. In the case of a hyperbola, the eccentricity \( e \) is greater than 1, distinguishing it from other conics.
For a given polar equation, you can identify the eccentricity by comparing and rewriting the equation into the form \( r = \frac{ed}{1 + e \cos \theta} \). Here, \( e \) determines the nature and shape of the curve:

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

This parameter not only determines the type of conic but also impacts the directrix and orientation of the curve.

Vertices

Vertices in the context of conics are specific points on the curve that are closest or farthest from the center or focus. For a hyperbola, the vertices are the points where each branch of the hyperbola is closest to the center.
When using polar coordinates, the vertices are calculated from the formula \( ed = 6 \) as shown in this example. The distance \( d \) from the pole to the nearest vertex is given by \( d = \frac{12}{7} \). This point is crucial as it marks one end of the hyperbola's branch and helps in sketching the curve.
In the plot, the vertices serve as starting points to draw each symmetric curve of the hyperbola, forming its distinctive open shape.

Directrix

The directrix is an essential, yet sometimes overlooked, element in the study of conic sections. It is a line that helps describe the conic's properties and its distance helps define the eccentricity along with the focus. For hyperbolas, the directrix is parallel to one axis of symmetry.
In polar coordinates, the directrix is located at a distance \( \frac{d}{e} \) from the pole. For the hyperbola in the given example, this distance is \( \frac{24}{49} \). This measurement is derived from the intersection of the directrix line and the shape's axis, delineating where the conic section broadens from.

• It is important to remember that the directrix aids in maintaining the relationship between the focus and the conic's eccentricity.
• It serves as a reference line, crucial in graphing and validating the shape of the hyperbola.

Recognizing and using the directrix appropriately ensures a comprehensive understanding of how the hyperbola behaves in its polar form.