Problem 55 - College Algebra

Question

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-3 ; e=\frac{1}{3}\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{3}{3 + \cos(\theta)} \).

Step 1: Understanding the problem

We need to find the polar equation of a conic section with the given directrix and eccentricity. The focus is at the origin, which is typical for polar equations.

Step 2: Identify relationship between polar and Cartesian

We start with the relationship of the directrix to the focus. The given directrix is a vertical line at \(x=-3\). For a vertical directrix in polar coordinates, the formula that relates these is: \[ r = \frac{ep}{1 + e\cos(\theta)} \] where \(e\) is the eccentricity and \(p\) is the distance from the focus to the directrix.

Step 3: Find the value of \(p\)

The value \(p\) is the distance from the focus at the origin to the directrix. Since the directrix is at \(x = -3\), the distance \(p\) is 3.

Step 4: Plug values into the polar equation

We plug in the given \(e = \frac{1}{3}\) and \(p = 3\) into the polar equation: \[ r = \frac{(\frac{1}{3})(3)}{1 + \frac{1}{3}\cos(\theta)} = \frac{1}{1 + \frac{1}{3}\cos(\theta)} \]

Step 5: Simplify the equation

To simplify, multiply the numerator and the denominator by 3 to eliminate the fraction within the denominator: \[ r = \frac{3}{3 + \cos(\theta)} \] Hence, the polar equation becomes: \( r = \frac{3}{3 + \cos(\theta)} \).

Key Concepts

Conic Sections

Conic sections are curves obtained by the intersection of a plane with a double-napped cone. These shapes are classified into four basic types: circles, ellipses, parabolas, and hyperbolas. The nature of the conic section depends on the angle of the plane with respect to the cone's axis.

• Circle: A special case of an ellipse where the eccentricity is zero.
• Ellipse: Features an eccentricity between 0 and 1 and appears as an elongated circle.
• Parabola: When the plane is parallel to the cone's edge, resulting in an eccentricity of 1.
• Hyperbola: Formed when the plane intersects both nappes, with an eccentricity greater than 1.

Understanding these shapes and their properties is crucial for visualizing and writing their equations in both Cartesian and polar forms.

Eccentricity

Eccentricity (denoted by \( e \)) is a crucial parameter that defines the shape of a conic section. It indicates how much a conic section deviates from being circular.

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

In the given exercise, the eccentricity is \( \frac{1}{3} \), indicating that the conic is an ellipse. This value helps us determine the conic's equations and visualize its shape, either stretched along the vertical or horizontal axis.

Directrix

The directrix of a conic section is an imaginary line that, along with the focus, is used to define the conic. It plays a significant role in the geometric definition of the conic and influences the polar equation.
For a conic with a focus at the origin, the distance \( p \) from the focus to the directrix is pivotal. In a polar coordinate system, the conic's equation involves the directrix directly as seen in formulas such as:
\[ r = \frac{ep}{1 + e\cos(\theta)} \]In this exercise, the directrix was a vertical line at \( x = -3 \). This helped us find \( p = 3 \) (distance from the origin to the directrix). The directrix, combined with eccentricity, aids in defining the conic's exact shape and position in the plane.