Problem 27 - College Algebra

Question

Convert the polar equation of a conic section to a rectangular equation. $$ r(2-\cos \theta)=1 $$

Step-by-Step Solution

Answer

The rectangular equation is more complex due to polar conversion.

Step 1: Simplify the Polar Equation

The given polar equation is \( r(2 - \cos \theta) = 1 \). First, solve for \( r \) by dividing both sides by \( 2 - \cos \theta \):\[ r = \frac{1}{2 - \cos \theta} \].

Step 2: Convert Cosine to Rectangular Form

Recall the relationship \( x = r\cos\theta \) and \( y = r\sin\theta \). To express \( \cos\theta \) in terms of \( x \) and \( y \), use \( \cos\theta = \frac{x}{r} \). Thus, the equation \( \cos\theta = \frac{x}{r} \) becomes \( r = \frac{x}{\cos\theta} \).

Step 3: Substitute and Rearrange

Substitute \( \cos \theta = \frac{x}{r} \) into the polar equation, yields \[ r = \frac{1}{2 - \frac{x}{r}} \]. Multiply through by \( r \) to clear the fraction: \( r^2 = r - \frac{x}{2} \).

Step 4: Express in Terms of \( x \) and \( y \)

Now, express \( r^2 \) as \( x^2 + y^2 \) (since \( r = \sqrt{x^2 + y^2} \)): \[ x^2 + y^2 = r - \frac{x}{2} \].

Step 5: Solve for \( r \)

Substitute \( r = \sqrt{x^2 + y^2} \) into the equation: \[ x^2 + y^2 = \sqrt{x^2 + y^2} - \frac{x}{2} \].

Step 6: Square Both Sides

Square both sides to eliminate the square root: \[ (x^2 + y^2)^2 = (\sqrt{x^2 + y^2} - \frac{x}{2})^2 \]. This expands to \[ (x^2 + y^2)^2 = x^2 + y^2 - x\sqrt{x^2 + y^2} + \frac{x^2}{4} \].

Step 7: Simplify the Equation

Gather all terms involving \( x^2 \) and \( y^2 \) on one side. Simplify the equation further to its simplest form by combining like terms.

Key Concepts

Polar Equations

Polar equations describe curves on a coordinate plane using a system of coordinates based on the radius and angle from the polar origin. In polar coordinates, any point is represented as \(r, \theta\), where \(r\) is the distance from the origin and \(\theta\) is the angle measured counterclockwise from the positive x-axis.
These equations are particularly useful for situations where the relationship between points relies on their distance from a specific origin, such as circles or other conic sections. For instance, circles centered at the origin have a simple polar equation: \(r = a\) where \(a\) is the radius.
In the example provided, the equation \(r(2 - \cos \theta) = 1\) is an instance of translating a conic section into polar form. Simplifying polar equations demands familiarity with trigonometric identities and an understanding of how changing \(\theta\) impacts \(r\).

Rectangular Equations

Rectangular equations, often referred to as Cartesian equations, use the familiar \(x\) and \(y\) coordinate system to describe curves. This form is more intuitive for expressing most geometric shapes and lines due to its familiarity from algebra.
The goal of converting a polar equation to a rectangular one, as seen in the original problem, is to translate the angular and radial descriptions into terms of horizontal and vertical distances from the coordinate plane origin. In other words, you replace \(r\) and \(\theta\) with \(x\) and \(y\) using known geometric relationships.
For example, \(x = r \cos \theta\) and \(y = r \sin \theta\) help bridge the gap between polar and rectangular descriptions, making it easier to apply algebraic methods to solve or analyze these equations.

Coordinate Conversion

Converting between polar and rectangular coordinates involves using trigonometric relationships. This conversion is essential when you need to apply polar equations to problems in a rectangular format, as it offers a different perspective to approach geometry related challenges.
When transforming a polar equation to a rectangular one, we typically use equations: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r = \sqrt{x^2 + y^2}\). Moreover, angular terms transform via trigonometric relations like \(\cos \theta = \frac{x}{r}\).
Using these relationships helps express equations like the example \(r(2 - \cos \theta) = 1\) in a more traditional algebraic format. This involves manipulating terms to isolate and replace \(r\) and \(\theta\) with \(x\) and \(y\), often involving additional algebraic steps such as squaring both sides or combining terms.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of involved variables. Understanding these identities is crucial when working with polar equations and converting them into rectangular form.
Key identities used in coordinate conversion include the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\) and the definitions of sine and cosine in terms of unit circle coordinates. For conversions, \(\sin \theta = \frac{y}{r}\) and \(\cos \theta = \frac{x}{r}\) are particularly significant.
These identities allow you to translate angular components into algebraic expressions involving \(x\) and \(y\). Additionally, they help simplify expressions and solve for unknowns in both polar and rectangular coordinate systems, as seen in transforming the given polar equation to its rectangular form.