Problem 30 - Precalculus: Functions and Graphs

Question

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=4, \quad r=-3 \csc \theta$$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{12}{1 + 4\sin\theta} \).

Step 1: Identify the conic type

The given eccentricity is \( e = 4 \). Since \( e > 1 \), the conic is a hyperbola.

Step 2: Recognize the directrix form

The given directrix equation is \( r = -3 \csc \theta \). This equation is similar to the standard form \( r = d \csc \theta \) for directrices, which suggests the directrix is a horizontal line, \( y = -3 \).

Step 3: Determine directrix distance and parameter

The distance \( d \) from the pole to the directrix is \( 3 \). In polar coordinates and for conics with focus at the pole, parameter \( p \) aligns with \( ed = p \).

Step 4: Calculate parameter \( p \)

Using \( p = ed \), substitute \( e = 4 \) and \( d = 3 \) to find \( p = 4 \times 3 = 12 \).

Step 5: Write the polar equation

The general polar equation for a conic with focus at the pole is \( r = \frac{ep}{1 - e\sin\theta} \) or \( r = \frac{ep}{1 + e\sin\theta} \). For the given directrix \( y = -3 \), use the negative sign. Substitute \( e = 4 \) and \( p = 12 \) to get: \[ r = \frac{12}{1 + 4\sin\theta} \].

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in a plane using a distance and an angle. Unlike Cartesian coordinates, which use x and y coordinates to define location, polar coordinates use a radius and an angle. The radius, denoted by \( r \), is the distance from a point to the origin (or pole), while the angle, denoted by \( \theta \), measures the rotation from the positive x-axis in a counterclockwise direction. This system is particularly useful for describing curves that have symmetries around a point, such as conic sections.

To convert from polar to Cartesian coordinates, you can use the formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

Conversely, converting Cartesian to polar involves:

• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \tan^{-1}(y/x) \)

Polar coordinates are particularly well-suited to problems involving polar equations of conics, where the conic's focus is at the pole.

Eccentricity

Eccentricity (denoted as \( e \)) is a key parameter in the study of conic sections. It defines the shape of a conic and its degree of deviation from being a circle. Each type of conic section has its characteristic eccentricity value, which can tell you immediately what type it is:

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

Eccentricity essentially measures how "stretched" a conic section is. A higher eccentricity in a hyperbola, for instance, means the branches are further apart, indicating a greater divergence from a circular shape.

In the given exercise, the eccentricity \( e = 4 \) classifies the conic as a hyperbola, as it exceeds 1.

Directrix

The directrix is an important line associated with a conic section. In the case of conics represented in polar coordinates, the directrix can often simplify to a specific line that guides the construction of the conic.

For hyperbolas in polar coordinates with the focus at the pole, the directrix typically takes a form like \( r = d \csc \theta \) or \( r = d \sec \theta \), depending on orientation. By setting up these formulas, the directrix helps maintain the proportional relationship between the distances from any point on the conic to the focus and the directrix line.

In this exercise, the directrix is given by \( r = -3 \csc \theta \), implying the line is horizontal and located at \( y = -3 \). This guides the creation of the appropriate polar equation.

Hyperbola

A hyperbola is a specific type of conic section characterized by its twin branches, which extend infinitely away from each other. This formation arises when a plane cuts through both nappes of a double cone. The hyperbola's distinct shape is described by its eccentricity \( e > 1 \). In addition, hyperbolas have properties such as asymptotes, which are straight lines that the hyperbola approaches but never intersects.

The general polar equation for a hyperbola with its focus at the pole is given as:

• \( r = \frac{ep}{1 - e\sin \theta} \) or \( r = \frac{ep}{1 + e\sin \theta} \)

These equations use the parameter \( p \), calculated in the given exercise as \( p = 12 \), determining the hyperbola's shape based on its directrix and eccentricity.

For the directrix \( y = -3 \) and eccentricity \( e = 4 \), the polar equation becomes: \[ r = \frac{12}{1 + 4\sin \theta} \]This describes how the radius changes with the angle, showing the hyperbola's distinct and stretched nature in the plane.