Problem 35 - Precalculus

Question

Use a graphing utility to graph the rotated conic. $$r=\frac{3}{1-\cos (\theta-\pi / 4)}$$

Step-by-Step Solution

Answer

The given conic section is a hyperbola with an eccentricity of 3, which is rotated 45° counterclockwise.

Step 1: Identify the eccentricity and rotation

In the given equation, \(r=\frac{3}{1-\cos (\theta-\pi / 4)}\), we can see that \(e=3\) and \(\phi =\pi / 4\). Hence, the eccentricity is 3, indicating that the conic section is a hyperbola, and it is rotated by \(\pi / 4\) radians or 45°.

Step 2: Create a table of values

To graph the polar equation, create a table of values for different angles \(\theta\). Select angles around the rotation \(\pi / 4\), record the value of \(r\) for each angle, taking care of the sign of \(r\).

Step 3: Plot the values

Use the values from the table to plot points in polar coordinates on a graphing utility. The points should form the shape of a hyperbola.

Step 4: Sketch the hyperbola

Connect the plotted points to draw the graph of the hyperbola. Make sure to sketch the branches of the hyperbola pointing in the direction where \(r\) increases from 0 as \(\theta\) increases from \(-\pi\) to \(\pi\).

Key Concepts

Eccentricity

Eccentricity is a crucial concept when analyzing conic sections. It helps determine the shape and nature of the conic. In mathematical terms, eccentricity, denoted by the symbol \( e \), is a number that describes how much a conic section deviates from being circular.

For a circle, the eccentricity is 0. For an ellipse, it's between 0 and 1. A hyperbola, like in the current exercise, has an eccentricity greater than 1. More specifically, here, the eccentricity is \( e = 3 \), signifying a hyperbola.

Understanding eccentricity allows us to categorize conics as ellipses, parabolas, or hyperbolas and gives insight into their geometric properties. When eccentricity increases, the conic section gets more "stretched".

Polar Coordinates

Polar coordinates are a unique way of locating points on a plane. Unlike the Cartesian system that uses a grid of \( x \) and \( y \) values, polar coordinates use a combination of an angle and a radius.

A point in polar coordinates is expressed as \( (r, \theta) \), where \( r \) is the radius – the distance from the origin to the point. The angle \( \theta \) is measured from the positive x-axis in a counter-clockwise direction.

This method is invaluable when dealing with curves or rotations, such as those seen in our rotated conic section example. To convert back and forth between Cartesian and polar, use the relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \tan^{-1}(y/x) \)

Hyperbola

A hyperbola is one of the fundamental conic sections, characterized by its two distinct curves or branches, which are symmetric and open away from each other. It arises when the plane intersects both nappes (the upper and lower portions) of a double cone.

In our exercise, the hyperbola is represented in the polar form by the equation \( r = \frac{3}{1-\cos(\theta-\pi/4)} \). The features of a hyperbola, such as asymptotes, foci, and vertices, can be derived from its standard equation.

Graphically, the hyperbola is often seen where the distance of any point on the curves from two fixed points (known as foci) is constant. A hyperbola's branches continue infinitely and are mirrored images about this central axis. Observing the hyperbola in polar coordinates enables the study of these properties in different orientations.

Graphing Utility

A graphing utility is a tool, either physical or digital, that helps visualize mathematical equations, allowing users to see the shape and form of the mathematical concepts they are studying.

In this exercise, utilizing a graphing utility to plot the points given by \( r = \frac{3}{1-\cos(\theta-\pi/4)} \) in polar coordinates provides a clear visual representation of the hyperbola. It simplifies the task of plotting and helps in sketching the final graph more accurately.

Modern graphing utilities include calculators, apps, and software like Desmos or GeoGebra. These tools not only allow plotting but can also manipulate the view, adjust scales, and even animate changes in parameters like rotation or eccentricity, enhancing understanding.