Problem 5 - Algebra and Trigonometry

Question

1–8 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 of the hyperbola is \( r = \frac{20}{1 + 4 \cos \theta} \).

Step 1: Understand the characteristics of a hyperbola

A hyperbola has an eccentricity greater than 1. Given that the eccentricity is 4, we know that we are dealing with a hyperbola since the eccentricity is greater than 1.

Step 2: Write the polar form equation

The polar form of a conic section with eccentricity \( e \) and directrix \( d \) is \( r = \frac{ed}{1 + e\cos\theta} \) if the directrix is vertical or \( r = \frac{ed}{1 + e\sin\theta} \) if the directrix is horizontal.

Step 3: Identify the type of directrix

The directrix given is \( r = 5 \sec\theta \), which is a vertical line. This suggests that \( \cos\theta \) is used in the equation.

Step 4: Substitute the values into the polar form equation

Since the directrix \( r = 5 \sec\theta \) corresponds to \( x = 5 \) in Cartesian coordinates, the distance \( d \) from the origin to this line is 5. Substitute \( e = 4 \) and \( d = 5 \) into the polar equation: \[ r = \frac{4 \cdot 5}{1 + 4\cos\theta} \].

Step 5: Simplify the equation

Simplify the equation: \[ r = \frac{20}{1 + 4 \cos \theta} \]. This is the equation of the hyperbola in polar coordinates with focus at the origin, eccentricity 4, and a vertical directrix.

Key Concepts

Polar Equations

Polar equations are a way to represent curves using a coordinate system based on angles and distances from a fixed point. Unlike the Cartesian coordinate system, which uses x and y axes, polar coordinates describe the position of a point as

• a distance ( ") from the origin (also known as the pole)
• an angle ( heta), measured from a reference direction

This system is particularly useful for dealing with conic sections that have their focus at the origin.
A conic section, such as an ellipse, parabola, or hyperbola, can be represented using a polar equation that depends on its eccentricity and directrix. The general polar form of a conic with its focus at the origin is written as:\[ r = \frac{ed}{1 + e \cos \theta} \] if the directrix is vertical, or as:\[ r = \frac{ed}{1 + e \sin \theta} \] if the directrix is horizontal.
This setup allows you to model conics like hyperbolas in a simple and intuitive way, making it easier to study their properties in polar coordinates.

Hyperbola

A hyperbola is one of the distinct types of conic sections formed when a plane cuts through a double cone. It has two separate curves, which may look like mirror images of each other.When represented in a polar coordinate system, it assumes a unique form. The equation of a hyperbola in polar form involves the eccentricity \( e \), which is greater than 1 for hyperbolas, and the directrix distance \( d \).Key features of a hyperbola include:

• Two distinct branches, each asymptotically approaching a pair of lines.
• An eccentricity \( e > 1 \).
• A focal length that extends to infinity, as the branches continue indefinitely.

In the problem at hand, the hyperbola satisfies these conditions with an eccentricity of 4 and vertical directrix. Using the directrix \( x = 5 \), the polar equation becomes:\[ r = \frac{20}{1 + 4 \cos \theta} \]This equation fully describes the curve's properties in relation to the pole (origin) and provides an easy way to visualize the hyperbola's shape and positioning in polar coordinates.

Eccentricity

Eccentricity is a crucial parameter in defining the shape of a conic section. It determines how "stretched" or "flattened" the conic appears.Here's how eccentricity values classify different conic shapes:

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

The exercise focuses on a hyperbola with an eccentricity of 4. This value means the hyperbola is significantly elongated compared to a circle or an ellipse.

Eccentricity measures how much the conic deviates from being circular. In a polar equation, eccentricity directly influences the curve’s equation form and its characteristics. For hyperbolas, larger eccentricity results in more spread-out branches. Understanding eccentricity helps not only in solving equations but also in visualizing the shapes and behaviors of the conic sections.