Problem 49 - College Algebra

Question

For the following exercises, 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 \( r = \frac{\frac{7}{8}}{1 + \frac{7}{2} \cos(\theta)} \).

Step 1: Identify the type of conic

The eccentricity value \(e = \frac{7}{2} > 1\) indicates that the conic is a hyperbola. If \(e > 1\), the conic is a hyperbola.

Step 2: Understand the polar equation format for conics

In polar coordinates, the equation of a conic section with focus at the origin is given by: \( r = \frac{ed}{1 + e \cos(\theta)} \) for a directrix \(x = d\). Here, the conic is opening towards the left since \(\cos(\theta)\) is used and the directrix is \(x = -\frac{1}{4}\).

Step 3: Transform the directrix

For the directrix \(x = -\frac{1}{4}\), we use \(d = \frac{1}{4}\) in our formula because the derivation formula for the polar conic involves using the absolute value of the directrix position, hence \(d = \left| -\frac{1}{4} \right| = \frac{1}{4}\).

Step 4: Plug values into the formula

Using \(e = \frac{7}{2}\) and \(d = \frac{1}{4}\), substitute into the polar equation: \[ r = \frac{\frac{7}{2} \cdot \frac{1}{4}}{1 + \frac{7}{2} \cos(\theta)} \]

Step 5: Simplify the polar equation

Simplify the expression:\[ r = \frac{\frac{7}{8}}{1 + \frac{7}{2} \cos(\theta)} \]Now, the polar equation of the hyperbola with the given parameters is:\[ r = \frac{7/8}{1 + \frac{7}{2} \cos(\theta)} \]

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They are a fundamental concept in geometry and appear in various mathematical contexts. The primary types of conics are:

• Ellipse
• Parabola
• Hyperbola

These curves have unique properties based on their eccentricity, a key parameter that determines their shape. In polar coordinates, these sections can be represented using a focus and a directrix, providing a different perspective compared to Cartesian coordinates. This helps in solving problems related to satellites, orbits, and the paths of celestial objects.

Hyperbola

A hyperbola is one of the fundamental types of conic sections characterized by having two identical branches that mirror each other. In simple terms, a hyperbola is formed when a plane cuts through both nappes of a cone. The key features of a hyperbola include:

• Two separate curves called "branches."
• An eccentricity greater than 1, which distinguishes it from ellipses and parabolas.
• Asymptotes, which create a framework around which the branches form.

In polar coordinates, hyperbolas typically involve a focus at the origin, making it easier to describe their geometric properties. The branches of a hyperbola move away from each other, resembling the opening of an hourglass. Understanding hyperbolas is crucial in fields like navigation and physics.

Eccentricity

Eccentricity is a crucial concept in defining and differentiating conic sections. It measures the deviation of a conic from being circular. The eccentricity (\(e\)) value depends on the type of conic section, acting as a determinant of its shape:

• \(e = 1\) for parabolas.
• \(e < 1\) for ellipses.
• \(e > 1\) for hyperbolas.

In our given exercise, the eccentricity is \( e = \frac{7}{2} \), indicating the conic is a hyperbola. This high eccentricity means the branches of the hyperbola open wide apart. Eccentricity is integral in identifying the nature of conic sections and plays a role in the orbits of celestial objects.

Directrix

The directrix is a fixed line used in the description of a conic section. It acts as a reference line along with the eccentricity to define the conic in polar coordinates. Here are some important points about the directrix:

• It is perpendicular to the axis of symmetry of the conic.
• In polar equations, conics use the directrix to represent distance and focus properties of the curve.
• The directrix helps determine the position and orientation of the conic.

In the provided exercise, the directrix \( x = -\frac{1}{4} \) signifies a certain orientation and magnitude, guiding how the conic (hyperbola) is shaped. By using both the focus and directrix, we can precisely define and calculate the position of any point on the conic.