Problem 7 - Precalculus: Mathematics for Calculus

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: Identify the conic type and given information

We are given that the conic is a hyperbola. The eccentricity of the hyperbola is given as \( e = 4 \), and the equation of the directrix is \( r = 5 \sec \theta \).

Step 2: Understand polar equation for a conic

The general polar equation for a conic is \( r = \frac{ed}{1 + e \cos \theta} \) if the directrix is \( x = d \), and \( r = \frac{ed}{1 - e \cos \theta} \) if the directrix is \( x = -d \). Since the given directrix is \( r = 5 \sec \theta \), convert this to Cartesian by \( x = 5 \).

Step 3: Determine the equation form to use

Since the directrix is \( x = 5 \), we use the polar equation: \( r = \frac{ed}{1 - e \cos \theta} \). Given the direction of the directrix, it's actually \( r = 5 \sec \theta \) converted to \( x = 5 \), indicating the minus sign in the denominator.

Step 4: Substitute in the known values

From the converted form \( x = d = 5 \), we know that \( d = 5 \). Substitute \( e = 4 \) and \( d = 5 \) into the equation: \[ r = \frac{4 \cdot 5}{1 - 4 \cos \theta} = \frac{20}{1 - 4 \cos \theta} \].

Step 5: Write the final polar equation

The polar equation of the conic is \( r = \frac{20}{1 - 4 \cos \theta} \) which represents a hyperbola with the given eccentricity and directrix.

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane. They come in four types: circles, ellipses, parabolas, and hyperbolas. Each type of conic section has a unique characteristic and equation in both Cartesian and polar coordinates.

• **Circle:** All points are equidistant from a central point.
• **Ellipse:** The sum of the distances from any point on the curve to two fixed points, called foci, is constant.
• **Parabola:** Every point is equidistant to a fixed point (focus) and a fixed line (directrix).
• **Hyperbola:** The difference in distances from any point on the curve to two foci is constant.

These sections are vital in various fields, such as physics, engineering, and astronomy. Understanding conic sections helps in modeling planetary orbits, designing reflective surfaces, and analyzing motion. The symmetry and shape dictated by the equations make these applications possible.

Hyperbola

A hyperbola is one of the conic sections formed when a plane cuts through both nappes of a cone. Its structure consists of two separate curves, known as branches. Unlike ellipses, hyperbolas have an intriguing property where the difference in distances to two fixed points is constant.
Hyperbolas can be expressed using a polar equation, especially useful when solving problems related to satellite dishes, navigation systems, and even in understanding certain astronomical phenomena.
The general form for a hyperbola's polar equation is derived based on its eccentricity and directrix:
- For hyperbolas, the directrix can influence whether the equation has "plus" or "minus" signs, changing how the curve opens relative to the coordinate axes.- The polar form \[ r = \frac{ed}{1 - e \cos \theta} \] is used when the directrix is of the form \( x = d \) and the focus is at the origin. Understanding hyperbolas helps in grasping complex systems in both mathematics and real-world applications.

Eccentricity

Eccentricity is a parameter associated with conic sections that defines their shape. It's denoted by the symbol \(e\) and is crucial in determining the type of conic section you have.

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

When eccentricity is greater than 1, the conic is a hyperbola. The greater the eccentricity, the more "stretched out" the hyperbola is. In the given problem, the eccentricity \(e = 4\) shows that the hyperbola is quite elongated.
Eccentricity ties into the equations we use to represent conics in polar coordinates. Knowing the eccentricity and the directrix allows us to write accurate polar equations that model the relationships and characteristics of these figures.