Problem 38 - Precalculus

Question

Use a graphing utility to graph the rotated conic. $$r=\frac{5}{-1+2 \cos (\theta+2 \pi / 3)}$$

Step-by-Step Solution

Answer

The conic plotted will be an ellipse rotated by \(2 \pi / 3\) radians due to the presence of \(\theta + 2 \pi / 3 \) in the given equation. Due to the nature of the equation, the focus of this ellipse isn't at the origin, leading to the rotation.

Step 1: Rewrite the equation in its rectangular form

The rectangular form for polar coordinates (\(r, \theta\)) is given by: \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \). Let's use these conversions to rewrite the polar equation in rectangular form. Plug the polar coordinates to \( r = \frac{5}{-1+2 \cos (\theta+2 \pi / 3)} \) to get \[ x = \frac{5 \cos(\theta)}{-1+2 \cos (\theta+2 \pi / 3)} \] and \[ y = \frac{5 \sin(\theta)}{-1+2 \cos (\theta+2 \pi / 3)} \]

Step 2: Analyze the rotation of the conic

Given the form of our polar equation and our translations to rectangular form, we can understand that the conic has been rotated by \(2\pi/3\) radians about the origin. The factor of \(2 \cos (\theta+2 \pi / 3)\) in the denominator indicates this rotation.

Step 3: Graph the conic

With the rectangular forms of \(x\) and \(y\), and understanding the rotation, plot the conic in a graphing utility. Input the expressions for \(x\) and \(y\) as parametric equations where \(\theta\) acts as the parameter. Since \(\theta\) is in polar coordinates ranging from \(0\) to \(2\pi\), ensure this range is accounted for when plotting.

Key Concepts

Polar to Rectangular Coordinates

When working with conics in polar form, one common task is to transform them into rectangular coordinates. This process helps in visualizing and analyzing the graph in the familiar Cartesian plane. To convert from polar coordinates

• where \( r \) is the distance from the origin to the point, and \( \theta \) is the angle from the positive x-axis,
• we use the formulas \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \).

Substituting the given polar equation \[ r = \frac{5}{-1 + 2 \cos(\theta + 2\pi/3)} \]into these formulas allows us to express the conic in terms of \( x \) and \( y \).
For example, we derive the rectangular equations:

• \( x = \frac{5 \cos(\theta)}{-1 + 2 \cos(\theta + 2\pi/3)} \)
• \( y = \frac{5 \sin(\theta)}{-1 + 2 \cos(\theta + 2\pi/3)} \).

This conversion is essential for further analysis and graphing.

Parametric Equations

Once we have the rectangular coordinates, we can express the relationship as parametric equations. This approach is especially helpful for graphing, as it easily allows us to map out the curve.A parametric equation expresses the coordinates \(x\) and \(y\) as functions of a third parameter. Here, \(\theta\) serves as this parameter, originating from the polar representation of the conic.

• Parametric form for \(x\) is: \( x(\theta) = \frac{5 \cos(\theta)}{-1 + 2 \cos(\theta + 2\pi/3)} \)
• Parametric form for \(y\) is: \( y(\theta) = \frac{5 \sin(\theta)}{-1 + 2 \cos(\theta + 2\pi/3)} \).

These expressions dictate the \(x\) and \(y\) coordinates for each angle \(\theta\) ranging typically from \(0\) to \(2\pi\).
Using parametric equations allows graphing utilities to plot the curve efficiently, as they're designed to handle expressions of the form \(x(t)\) and \(y(t)\).

Graphing Conics

Graphing conics, especially when they are rotated, can be challenging but rewarding. Using a graphing utility, we input our parametric equations directly.
These utilities can handle the complexity of conics expressed in terms of a parameter like \(\theta\) used here.To successfully graph our conic:

• Make sure to set the parameter range for \(\theta\) from \(0\) to \(2\pi\) so that the entire conic section is captured.
• Input \(x(\theta)\) and \(y(\theta)\) into the graphing tool to plot points corresponding to each value of \(\theta\).

Once these points are plotted, the tool should connect them smoothly to provide the full image of the conic.
Sometimes, viewing the rotation helps in understanding the nature of the conic, such as ellipses or hyperbolas, and their orientation in the Cartesian plane.