Problem 29 - College Algebra

Question

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{6 \sec \theta}{-2+3 \sec \theta} $$

Step-by-Step Solution

Answer

The rectangular equation is \( x^3 + xy^2 = 6\).

Step 1: Identify and Arrange the Polar Equation

The given polar equation is \(r = \frac{6 \sec \theta}{-2+3 \sec \theta}\). Start by rearranging the equation to express \( r \) in terms of \( \theta \). In this case, the equation is already simplified in terms of \( \sec \theta \).

Step 2: Use the Relationship with Rectangular Coordinates

Recall the relationship between rectangular and polar coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Additionally, we use the identity \( \sec \theta = \frac{1}{\cos \theta} \). Substitute \( \sec \theta \) from the equation into \( r \cos \theta = x \).

Step 3: Solve for \( r \)

Substitute \( \sec \theta = \frac{1}{\cos \theta} \) into the equation: \[ r = \frac{6 \cdot \frac{1}{\cos \theta}}{-2 + 3 \cdot \frac{1}{\cos \theta}} \] which simplifies further to \(r \cos \theta = \frac{6}{3 - 2 \cos \theta}\). Multiply both terms by \( \cos \theta \) to clear the fraction.

Step 4: Express \( r \cos \theta \) in Terms of \( x \)

Since \( r \cos \theta = x \), replace \( r \cos \theta \) in the equation to get: \[ x = \frac{6}{3 - 2 \cos \theta} \].

Step 5: Replace \( \cos \theta \) with \( \frac{x}{r} \)

To convert \( \cos \theta \), use the relation \( \cos \theta = \frac{x}{r} \). Substitute \( \cos \theta \) in the equation: \[ x = \frac{6}{3 - 2 \frac{x}{r}} \]. Now clear the fraction by multiplying through by \(r\), giving \[ xr = 6 \rightarrow r(3r - 2x) = 6 \].

Step 6: Substitute \( r^2 = x^2 + y^2 \)

Utilize the relationship \( r^2 = x^2 + y^2 \), substitute \( r^2 \) in the equation: \( x(x^2 + y^2) = 6 \).

Step 7: Simplify the Equation

Distribute the \( x \) and rearrange the equation: \( x^3 + xy^2 = 6 \). This is the rectangular form of the given polar equation.

Key Concepts

Understanding Polar Equations

A polar equation represents a curve by expressing the coordinates in terms of angle \( \theta \) and the radial distance \( r \) from a fixed point, known as the pole. Exploring the polar equation, such as \( r = \frac{6 \sec \theta}{-2+3 \sec \theta} \), we find that components of these expressions are handy for describing certain curves. In this particular polar equation, choose \( \sec \theta \) to simplify expressions using trigonometric identities.
These types of equations are useful for representing shapes like conic sections, including ellipses and hyperbolas, with symmetry centered on the origin. When working with polar equations, understanding trigonometric identities can greatly aid in manipulating the expressions for converting purposes.

Conversion to Rectangular Equations

Rectangular equations utilize Cartesian coordinates \( x \) and \( y \) to describe the same curves as polar equations but in a coordinate plane. The conversion involves using relationships such as \( x = r \cos \theta \) and \( y = r \sin \theta \). Recognize that \( \sec \theta \) can be replaced using its relation \( \sec \theta = \frac{1}{\cos \theta} \).
In our example, substitute and simplify the equation to bring \( r \cos \theta \) in terms of \( x \). By doing so, you'll find an equation involving \( x \) and \( y \), ultimately reaching a form like \( x^3 + xy^2 = 6 \). Transformational techniques like these are crucial for analyzing different views of geometric figures.

Conic Sections and Their Polar Forms

Conic sections are curves obtained by intersecting a cone with a plane in various ways to produce ellipses, parabolas, and hyperbolas. These are elegantly described using polar coordinates with expressions that neatly incorporate angles and distances. The nature of the polar conic equation depends on constants in the formula. For instance, a polar equation in the form \( r = \frac{e}{1 + e \cos \theta} \) (where \( e \) is eccentricity) can represent different conics.

• \( e = 1 \) represents a parabola
• \( e < 1 \) represents an ellipse
• \( e > 1 \) denotes a hyperbola

In polar form, conic sections allow for an easy identification of properties like center and symmetry, which may be less apparent in rectangular form.

Coordinate Transformation Techniques

Coordinate transformation refers to the conversion of equations or coordinates from one system to another like from polar to rectangular coordinates. This process simplifies the problem-solving approach, especially in calculus and geometry.
Using relationships such as \( r^2 = x^2 + y^2 \) and \( \cos \theta = \frac{x}{r} \), you can transition between systems. This transformation aids in tackling problems in various contexts by providing different perspectives or simplifying calculations. In our example, you use transformations to make an equation manageable in analysis. Recognize how each system offers its strengths: polar is often simpler for rotationally symmetric problems, while rectangular is preferred for problems aligned along Cartesian axes. By mastering these techniques, you broaden your ability to work with different mathematics problems efficiently.