Problem 23 - College Algebra

Question

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{3}{8-8 \cos \theta} $$

Step-by-Step Solution

Answer

The rectangular equation is \( y^2 = \frac{3}{4}x + \frac{9}{64} \).

Step 1: Identify Conic Properties

Given the polar equation \( r = \frac{3}{8 - 8 \cos \theta} \), recognize that it is in the form \( r = \frac{ed}{1 - e\cos\theta} \), which represents a conic section with the focus at the origin. Here, \( e = 1 \) and \( d = 3 \). Since \( e = 1 \), this is a parabola.

Step 2: Recognize Coordinate Transformation

Recall the relationship between polar and rectangular coordinates: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r^2 = x^2 + y^2 \). Our goal is to express this equation in terms of \( x \) and \( y \).

Step 3: Eliminate \(\theta\) Using Trigonometric Identity

The given polar equation is \( r = \frac{3}{8 - 8 \cos \theta} \). Multiply both sides by the denominator to get \( r(8 - 8 \cos \theta) = 3 \). Simplify to find \( 8r - 8r \cos \theta = 3 \). Recall that \( r \cos \theta = x \), so substituting gives \( 8r - 8x = 3 \).

Step 4: Solve for \( r \)

Rearrange the equation to solve for \( r \): \( 8r = 8x + 3 \), thus \( r = x + \frac{3}{8} \).

Step 5: Substitute \( r \) and Solve for an Equation in \( x \) and \( y \)

Use \( r = \sqrt{x^2 + y^2} \) to substitute for \( r \): \( \sqrt{x^2 + y^2} = x + \frac{3}{8} \). To eliminate the square root, square both sides to get \( x^2 + y^2 = \left(x + \frac{3}{8}\right)^2 \). Expand and simplify to find the rectangular equation.

Step 6: Simplify the Rectangular Equation

Expanding the right-hand side, we have \( x^2 + y^2 = x^2 + \frac{3}{4}x + \frac{9}{64} \). Subtract \( x^2 \) from both sides to simplify: \( y^2 = \frac{3}{4}x + \frac{9}{64} \).

Key Concepts

Coordinate Transformation

When dealing with mathematical equations, sometimes it is necessary to switch between different systems of coordinates. One of the most common transformations is from polar coordinates to rectangular coordinates. Polar coordinates describe a point in terms of its distance from the origin and its angle from a reference direction. This means any point can be expressed as \( r, \theta \). Conversely, rectangular coordinates provide a point using horizontal and vertical distances from the origin, often denoted as \( x, y \). The transformation equations are:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r^2 = x^2 + y^2 \)

Use these relationships as a bridge to switch from polar to rectangular coordinates, which simplifies analyzing motion or shapes within various mathematical and physical problems.

Conic Section in Polar Coordinates

Conic sections are curves obtained by intersecting a cone with a plane. In polar coordinates, these curves can be represented by equations of the form \( r = \frac{ed}{1 - e\cos\theta} \), where \( e \) (the eccentricity) and \( d \) are constants.Eccentricity \( e \) determines the type of conic:

• \( e < 1 \): Ellipse
• \( e = 1 \): Parabola
• \( e > 1 \): Hyperbola

In the exercise, the given equation \( r = \frac{3}{8 - 8 \cos \theta} \) showed that \( e = 1 \), indicating it is a parabola. Recognizing the conic section helps us understand its properties and behavior, like the shape and orientation.

Rectangular Equation of Conic

To better analyze a conic section, sometimes it is necessary to convert its polar equation to a rectangular form. This rectangular equation uses the familiar \( x, y \) plane, which helps in plotting and understanding the conic's shape. For the given polar equation, the rectangular coordinates allow us to remove the angle \( \theta \) by substituting \( r \cos \theta = x \) and \( r \sin \theta = y \). After these substitutions, solve the equation algebraically to express it purely in terms of \( x \) and \( y \). This process simplifies understanding and solving geometry and calculus problems related to conic sections.

Polar Equation to Rectangular Form

Converting a polar equation to a rectangular form involves manipulating the original polar expression into one that involves only rectangular variables \( x \) and \( y \). This conversion is crucial when working within a cartesian coordinate system for easier visualization and analysis.Follow these main steps:

• Substitute the polar identity \( r = \sqrt{x^2 + y^2} \).
• Use \( x = r \cos \theta \) and \( y = r \sin \theta \) to eliminate trig functions.
• Square both sides if necessary to remove square roots or isolate a particular term.

For example, with the equation \( r = \frac{3}{8 - 8 \cos \theta} \), we eventually simplified to an equation defining \( y^2 = \frac{3}{4}x + \frac{9}{64} \). These steps yield a resultant rectangular equation that's easier for graphing and further mathematical exploration.