Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{2}{5-3 \sin \theta}\)

Step-by-Step Solution

Answer

The rectangular equation involves simplifying \( 5(x^2 + y^2) - 3y\sqrt{x^2+y^2} = 2 \).

Step 1: Identify the polar equation format

The given polar equation is \( r = \frac{2}{5 - 3 \sin \theta} \). Recognize this as the standard form for a conic section equation with polar coordinates, which may take the form \( r = \frac{ed}{1 + e \sin \theta} \) or \( r = \frac{ed}{1 + e \cos \theta} \). From this equation, we see it can fit \( r = \frac{ed}{1 - e \sin \theta} \).

Step 2: Identify components

By comparing \( r = \frac{2}{5 - 3 \sin \theta} \) with the standard polar conic form \( r = \frac{ed}{1 - e \sin \theta} \), we determine \( ed = 2 \), \( e = 3 \), and \( 5 \) is related to the denominator constant of \( 1 - e \sin \theta \). Therefore, applying \( r = \frac{2}{5 - 3 \sin \theta} \), we identify \( d = \frac{2}{3} \).

Step 3: Use polar to rectangular transformations

We know that if \( x = r \cos \theta \) and \( y = r \sin \theta \), then \( r = \sqrt{x^2 + y^2} \) and \( \sin \theta = \frac{y}{r} \). We will use these to convert to rectangular coordinates.

Step 4: Insert identities into polar equation

Insert \( r = \sqrt{x^2 + y^2} \) and \( \sin \theta = \frac{y}{r} \) into the equation \( r = \frac{2}{5 - 3 \sin \theta} \). This forms the equation \[ \sqrt{x^2 + y^2} = \frac{2}{5 - 3\left( \frac{y}{\sqrt{x^2 + y^2}} \right)} \].

Step 5: Clear the fraction

Multiply through by the denominator \( 5\sqrt{x^2 + y^2} - 3y \) to remove the fraction: \[ \sqrt{x^2 + y^2} (5\sqrt{x^2+y^2} - 3y) = 2 \].

Step 6: Simplify the equation

Distribute the \( \sqrt{x^2+y^2} \): \[ 5(x^2 + y^2) - 3y\sqrt{x^2+y^2} = 2 \].

Step 7: Solve for rectangular terms

Reorganize the equation to isolate \( y \): \( 3y\sqrt{x^2+y^2} = 5(x^2 + y^2) - 2 \). Notice that further simplification involves isolating variables completely if possible.

Step 8: Structural rectification

This yields no further algebraic simplification without specific \( y \) or function transformations. However, continue examining integer solutions and multiplicative inversions as \( \sqrt{x^2 + y^2} \) can continue to denote deeper core simplifications in practice.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. The main types of conic sections are circles, ellipses, parabolas, and hyperbolas. They play a key role in mathematics and geometry due to their unique properties and applications.
In polar coordinates, conic sections are often expressed in the form \( r = \frac{ed}{1 + e \sin \theta} \) or \( r = \frac{ed}{1 + e \cos \theta} \), where \( e \) is the eccentricity of the conic. This eccentricity determines the type of conic section:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), the conic is an ellipse.
• If \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), the conic is a hyperbola.

Understanding these forms helps transition between polar and rectangular equations.

Rectangular Equations

In mathematics, a rectangular equation is a representation of a mathematical relationship in the Cartesian coordinate system, which uses an \( x \)-\( y \) plane. For example, instead of using polar coordinates \( (r, \theta) \), rectangular coordinates use \( (x, y) \), focusing on horizontal and vertical axes.
This conversion involves replacing polar terms with their rectangular counterparts, such as using \( x = r \cos \theta \) and \( y = r \sin \theta \).
Rectangular equations are often more intuitive for plotting and solving algebraically, especially since they align directly with the familiar geometry of the Euclidean plane. They can be crucial as they simplify the process of analyzing and graphing conics using standard coordinate graphs.

Polar to Rectangular Conversion

Converting between polar and rectangular coordinates is essential for analyzing and understanding problems in different settings. Polar to rectangular conversion uses the relationships:

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

When given a polar equation, these formulas directly translate polar expressions like \( \sin \theta \) into rectangular terms such as \( \frac{y}{r} \). These conversions allow us to better visualize and solve equations by utilizing the Cartesian system, which is often more straightforward for algebraic manipulation and graphical interpretation.

Trigonometric Identities

Trigonometric identities are essential tools in converting and simplifying equations involving angles. They provide relationships among trigonometric functions that remain true for all values of the variables.
When converting polar equations to rectangular form, identities such as \( \sin^2 \theta + \cos^2 \theta = 1 \) play a crucial role in resolving expressions.
For our case, understanding that \( \sin \theta = \frac{y}{r} \) aids in transforming parts of a polar equation into forms calculable in rectangular coordinates.
This is key to mapping the unique paths of conic sections, ensuring streamlined problem-solving and enhancing comprehension of spatial relationships within different coordinate systems.