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 Elements for the Polar Equation

To write the polar equation of a conic, identify its type (hyperbola), eccentricity \(e = 4\), and directrix given by \(r = 5 \sec \theta\). A hyperbola has an eccentricity greater than 1.

Step 2: Use the Polar Equation Formula for Hyperbolas

The standard polar equation for a conic section with focus at the origin when the directrix is parallel to the polar axis is \(r = \frac{ed}{1 + e \cos \theta}\). Identify the correct transformation since we're given \(r = 5 \sec \theta\).

Step 3: Transform the Directrix

Rewrite the given directrix equation in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\), so \(r \cos \theta = 5\). Since \(r = \frac{ed}{1 + e \cos \theta}\), equate this to the condition and find \(d\).

Step 4: Calculate the Directrix Distance

From \(r \cos \theta = 5\), we have \(d = 5\). The directrix is thus \(d = 5\). Substitute into the equation \(r = \frac{ed}{1 + e \cos \theta}\).

Step 5: Substitute Known Values

Substitute \(e = 4\) and \(d = 5\) into the polar equation: \[ r = \frac{4 \cdot 5}{1 + 4 \cos \theta} \].

Step 6: Simplify the Polar Equation

Simplify to get the final polar equation: \[ r = \frac{20}{1 + 4 \cos \theta} \].

Key Concepts

Eccentricity

Eccentricity is a fundamental concept in understanding conic sections. It measures how "stretched" a conic section is. The eccentricity, denoted as \( e \), determines the shape and type of conic. For example:

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

In the given exercise, we have a hyperbola with an eccentricity of 4. Because the eccentricity is greater than 1, we know the conic section is indeed a hyperbola. Understanding eccentricity is crucial as it provides a quick and clear distinction between different conic shapes, allowing us to predict their properties and behaviors.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. Depending on the angle and location of the intersection, different conics are formed:

• Circle
• Ellipse
• Parabola
• Hyperbola

These conic sections have distinctive mathematical properties and polar equations. Each type of conic section represents a unique path in geometry, translating into various applications in physics, engineering, and astronomy. In our problem, the hyperbola is the conic section in focus. This type of conic is formed when the plane intersects both nappes of the cone, leading to two distinct, mirror-image branches.

Hyperbola

A hyperbola is a type of conic section characterized by its two separate curves, known as branches. In the context of polar coordinates, the hyperbola's defining equation reflects its eccentricity, which is greater than one. To model a hyperbola in polar coordinates, we use:

• Its focal point, located at the origin in our exercise.
• Its directrix and eccentricity, which together determine its unique parameters.
• The polar equation is: \[ r = \frac{ed}{1 + e \cos \theta} \]

This formula shows the relationship between each point \((r, \theta)\) on the hyperbola. In our case, with an eccentricity of 4 and a directrix of \(5 \, \sec \theta\), these parameters control the hyperbola's shape and position. As such, hyperbolas have important real-world applications, including satellite dish designs and certain types of orbits.

Directrix

The directrix is a crucial line used in defining conic sections. It works together with the focus and eccentricity to determine the position of each point on the conic. For a hyperbola, like in this exercise, the directrix is especially significant:- It provides a reference line, used to define the relationship between distances on the conic.- In polar coordinates, it helps form equations that describe the conic's geometry.For example, in the given exercise, the directrix is aligned with the polar axis as \(r = 5\sec\theta\). This equation can be transformed by substituting \(\sec\theta = \frac{1}{\cos\theta}\) into a more familiar form, giving us the distance formula \(r \cos\theta = 5\). Knowing the directrix allows us to solve and find the constants needed for the polar equation of our hyperbola. Essentially, the directrix acts as a guide in shaping the curve, contributing to the understanding and solving of conic sections problems.