Problem 7 - Algebra and Trigonometry

Question

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Hyperbola, eccentricity \(4,\) directrix \(r=5 \sec \theta\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{20}{1 - 4\cos(\theta)} \).

Step 1: Understand the Polar Equation of a Conic

The general form of the polar equation of a conic with its focus at the origin is \( r = \frac{ed}{1 - e\cos(\theta)} \), where \(e\) is the eccentricity, and \(d\) is the distance from the directrix to the origin. For a hyperbola, \(e > 1\).

Step 2: Identify Given Values

We are given that the eccentricity \(e = 4\) and the directrix equation is \( r = 5 \sec \theta \). This means that the distance \(d = 5\).

Step 3: Formulate the Polar Equation

Substitute \(e = 4\) and \(d = 5\) into the polar equation of a conic. This gives us \( r = \frac{4 \cdot 5}{1 - 4\cos(\theta)} = \frac{20}{1 - 4\cos(\theta)} \).

Step 4: Simplify the Polar Equation

Write the simplified form of the equation: \( r = \frac{20}{1 - 4\cos(\theta)} \).

Key Concepts

hyperbola

A hyperbola is a type of conic section that opens outward, forming two disconnected curves. Imagine the shape of two opposite-facing arches. Unlike a circle or ellipse, which are closed curves, a hyperbola is open. It is defined as the set of all points where the difference in distances to two fixed points (foci) is constant. This is different from an ellipse, where you consider the sum of distances.

• It consists of two branches.
• Each branch has a distinct vertex.
• A hyperbola's axes of symmetry are the lines through its foci and vertices.

One way to identify a hyperbola is by looking at its eccentricity. If the eccentricity (denoted as e) is greater than 1, then the conic is a hyperbola.
An important note about hyperbolas is that they have no real maximum or minimum points, as they extend infinitely. They are often used in navigation systems, such as GPS, because they represent paths traced by signals.

eccentricity

Eccentricity is a crucial number in the study of conic sections that tells you how much a conic deviates from being circular. It's a measure of how stretched out it is. Each type of conic has its own characteristic eccentricity:

• A circle has an eccentricity of 0.
• An ellipse has an eccentricity between 0 and 1.
• A parabola has an eccentricity equal to 1.
• A hyperbola has an eccentricity greater than 1.

In our problem, we have a hyperbola with an eccentricity of 4. This indicates that the hyperbola is significantly elongated. The greater the eccentricity, the "flatter" the hyperbola appears. Eccentricity is a ratio of distances: the distance from a point on the conic to the focus and the distance from the same point to the directrix.
Ultimately, eccentricity helps us classify conics and understand their geometric properties.

polar equation of a conic

The polar equation of a conic simplifies the process of representing conic sections, such as ellipses, parabolas, and hyperbolas, in polar form. For conics centered at the origin with a focus at the origin, the equation appears as:\[ r = \frac{ed}{1 - e\cos(\theta)} \]In this equation:

• r is the radial distance from the origin.
• e represents the eccentricity.
• d is the distance from the directrix.
• \(\theta\) is the polar angle.

This formula is particularly useful in understanding how conics behave in a polar coordinate system. In the provided solution, substituting the values for eccentricity and directrix distance yields the specific polar equation for our hyperbola.
This concise equation provides a clear representation of the conic's shape and orientation in the coordinate plane.

directrix

The directrix is a fixed line used in tandem with a focus to define a conic section. It helps in constructing the set of points that make up the conic:

• For a parabola, it's the line from which point distances are compared to form the parabola.
• For an ellipse or hyperbola, it works with eccentricity to determine the locus of points.

In our polar context, the directrix plays a vital role in constructing the conic's equation. For example, when you have a polar equation, the distance of the directrix from the origin ( d ) is a key component. With a hyperbola, as in this task, the directrix provides a base reference line for measuring eccentricity-related distances.
Understanding the role of the directrix helps clarify why conic sections have specific shapes. It provides a reference for distinguishing the positions and distances involved in formulating conics.