Problem 22 - Precalculus

Question

Identify the conic and sketch its graph. $$r=\frac{9}{3-2 \cos \theta}$$

Step-by-Step Solution

Answer

The given equation represents a hyperbola with the focus at the origin and the directrix at x=4.5. The hyperbola opens to the right of the directrix.

Step 1 - Identify the conic section

The standard form of a conic in polar coordinates is \(r=\frac{p}{1 \pm e cos \theta}\). Here 'e' is the eccentricity: for \(e<1\) it is an ellipse, for \(e=1\) it is a parabola, and for \(e>1\) it is a hyperbola. In our equation \(r=\frac{9}{3-2cos\theta}\), p=9 and e=2. Since 'e' is greater than 1, the given equation represents a hyperbola.

Step 2 - Determine the focus

For a hyperbola, the focus is always at the origin. Hence, our focus for this equation is at the origin, i.e., (0,0).

Step 3 - Identify directrix

For the given conic in polar coordinates, the directrix is given by \(x=\frac{p}{e}\). Substituting the given values, we have the directrix as \(x=\frac{9}{2}\), i.e., the vertical line at x=4.5.

Step 4 - Sketch the graph

First sketch the directrix (the line x=4.5), then mark the focus at the origin. Because the denominator of the original equation is \(3-2cos\theta\), the directrix is horizontal to the right. Thus, sketch the hyperbola opening to the right of the directrix, with the focus at the origin.

Key Concepts

Polar Coordinates

Polar coordinates provide an alternative way to specify locations in a plane using a distance and angle measure. Instead of using Cartesian coordinates (x, y), where each point is located by moving horizontally and vertically from an origin, polar coordinates use two values:

• \( r \): the radial distance from the origin to the point.
• \( \theta \): the angle from the positive x-axis to the line connecting the origin to the point.

This system is particularly helpful when dealing with curves and figures such as circles, spirals, and conic sections.
When working with conics in polar coordinates, equations are typically given in forms such as \[ r = \frac{p}{1 + e \cos \theta} \]Where the focus of the conic is at the origin. This representation is elegant for conics like ellipses, parabolas, and hyperbolas as it naturally incorporates eccentricity and orientation within the plane.

Eccentricity

Eccentricity is a key concept in the study of conic sections. It is a number that describes how much a conic section deviates from being circular.

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

Eccentricity helps to understand the shape and type of the conic.
In the given equation \( r = \frac{9}{3 - 2 \cos \theta} \), the eccentricity is 2, which indicates the conic is a hyperbola.
Eccentricity further dictates the geometry such as the directrix of the conic. A higher eccentricity means that the conic is more stretched along its axis of symmetry, showing more pronounced opening for hyperbolas.

Hyperbola

A hyperbola is one of the four types of conic sections, which are the results of slicing a double cone with a plane. A hyperbola appears as two mirrored curves, known as branches.

In the context of polar coordinates, the equation of a hyperbola can be written as:\[ r = \frac{p}{1 - e \cos \theta} \]Here, \( e > 1 \) identifies it as a hyperbola.

• The parameter \( p \) is related to the distance between the directrix and the focus.
• The focus is located at the origin (0,0).
• The directrix helps plot the orientation and shape of the hyperbola.

Given the example \( r = \frac{9}{3 - 2 \cos \theta} \), you have:

• Focus: (0,0), at the origin.
• Directrix: \( x = \frac{9}{2} \), which is the line \( x = 4.5 \).

By identifying these components, you can sketch the hyperbola with its branches opening to the right.