Problem 49 - College Algebra

Question

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-\frac{1}{4} ; e=\frac{7}{2}\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{7}{8 - 14\cos\theta} \).

Step 1: Understand the Conic Type

Since the eccentricity \(e = \frac{7}{2} > 1\), the conic is a hyperbola. Hyperbolas are defined by \(e > 1\).

Step 2: Polar Equation Format

The general formula for a conic in polar coordinates with the focus at the origin is \( r = \frac{ed}{1 + e\cos\theta } \) if the directrix is to the right of the origin. For this problem, we need to adjust the formula because the directrix is \(x = -\frac{1}{4}\).

Step 3: Adjust for Directrix Position

When the directrix is along \(x = -\frac{1}{4}\), it means it is to the left of the origin. The polar equation modifies to \( r = \frac{ed}{1 - e\cos\theta}\) as the directrix is left of the pole.

Step 4: Calculate 'd' from the Directrix

The directrix is given as \(x = -\frac{1}{4}\). Since \(d\) is the distance from the pole to the directrix along the x-axis, \(d = \frac{1}{4}\).

Step 5: Substitute Values into Polar Equation

Replace \(e\) and \(d\) into the equation: \( r = \frac{\left( \frac{7}{2} \right) \left( \frac{1}{4} \right) }{1 - \frac{7}{2}\cos\theta} = \frac{\frac{7}{8}}{1 - \frac{7}{2}\cos\theta} = \frac{7}{8 - 14\cos\theta} \)

Step 6: Simplify the Equation

The simplified polar equation of the conic is \(r = \frac{7}{8 - 14\cos\theta}\).

Key Concepts

Eccentricity

Eccentricity is a numerical parameter that describes the shape of a conic section. It plays an essential role in determining what type of conic we are dealing with. Eccentricity is represented by the letter \( e \). Depending on its value, it can indicate whether the conic section is a circle, an ellipse, a parabola, or a hyperbola.

• For a circle, \( e = 0 \).
• For an ellipse, \( 0 < e < 1 \).
• For a parabola, \( e = 1 \).
• For a hyperbola, \( e > 1 \).

In this exercise, we were given an eccentricity of \( e = \frac{7}{2} \). Since \( e > 1 \), it confirms that we are dealing with a hyperbola. Understanding eccentricity is crucial because it allows us to identify the type of conic without visual inspection.

Hyperbola

A hyperbola is one of the conic sections that can be defined in several ways. One common definition is based on the difference of distances from any point on the hyperbola to two fixed points called foci. This difference is constant. However, in polar coordinates, the hyperbola can be described using its eccentricity and directrix.

The hyperbola, being defined by \( e > 1 \), means it consists of two separate curves or branches that mirror each other.

• Hyperbolas have two axes of symmetry: a transverse axis and a conjugate axis.
• They approach asymptotes but never touch them.
• The polar equation for a hyperbola takes a specific form, which varies based on the placement of the directrix.

In polar coordinates, the generic equation of a hyperbola is modified depending on whether the directrix is to the left or right of the origin. Understanding these characteristics can help visualize and solve problems related to hyperbolas, just as in the exercise provided.

Directrix

The directrix is a fundamental component in defining conic sections. It is a fixed line used alongside the focus to determine the path of the conic. For the conics formed in polar coordinates, the directrix influences the form of the equation representing the conic.

In our exercise, the directrix is along \( x = -\frac{1}{4} \), indicating it is positioned to the left of the focus, which is at the origin. The directrix affects the polar equation by changing the sign in the denominator.

• If the directrix is to the right of the origin, the equation is \( r = \frac{ed}{1 + e\cos\theta} \).
• If it is to the left, it changes to \( r = \frac{ed}{1 - e\cos\theta} \).

For this particular problem, we used the latter form to correctly reflect the positioning of the directrix. By comprehending the role of the directrix, we can correctly set up and solve the conic's polar equation, ensuring accurate representations of conic sections in polar coordinates.