Problem 24 - College Algebra

Question

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{2}{6+7 \cos \theta} $$

Step-by-Step Solution

Answer

Final simplified rectangular equation: \(36(x^2+y^2) = (2 - 7x)^2\).

Step 1: Identify the Polar Equation

The given polar equation is \(r = \frac{2}{6 + 7 \cos \theta}\). This is the equation of a conic in polar coordinates where \(e = \frac{7}{6}\) and \(d = \frac{2}{6}\).

Step 2: Convert to Rectangular Coordinates

We use the polar to rectangular conversion formulas: \( r = \sqrt{x^2 + y^2} \), \( x = r \cos \theta \), and \( y = r \sin \theta \). Notice \(r\cos \theta = x\) which gives \(\cos \theta = \frac{x}{r}\).

Step 3: Replace \(r\) and \(\cos \theta\)

Substitute \(r = \sqrt{x^2 + y^2}\) and \(\cos \theta = \frac{x}{\sqrt{x^2 + y^2}}\) into the polar equation. We get \( \sqrt{x^2 + y^2} = \frac{2}{6 + 7 \frac{x}{\sqrt{x^2 + y^2}}} \).

Step 4: Simplify the Equation

To eliminate the denominator on the right side, multiply each side by the denominator: \((6 + 7\frac{x}{\sqrt{x^2 + y^2}})\sqrt{x^2 + y^2} = 2\). This implies \(6\sqrt{x^2 + y^2} + 7x = 2\).

Step 5: Further Simplification

Solve for \(\sqrt{x^2 + y^2}\) by isolating terms: \(6\sqrt{x^2 + y^2} = 2 - 7x\). Finally, square both sides to eliminate the square root.

Step 6: Final Simplified Rectangular Equation

The final equation by squaring is \(36x^2 + 36y^2 = (2 - 7x)^2\), after simplification this results in the rectangular form of the conic section.

Key Concepts

Polar Coordinates

Polar coordinates offer a different perspective from Cartesian coordinates. Instead of using x and y, polar coordinates express points with a radius and angle. These are denoted as \( (r, \theta) \), where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis. This system is especially useful for dealing with curves and spirals, often simplifying equations that seem complex in rectangular form.
In many cases, polar coordinates make trigonometric functions and rotations easier to understand and work with. Imagine drawing a circle: it’s easier to visualize the sweep of an angle rather than movements along x and y. Therein lies the power of polar coordinates for conic sections. It simplifies calculations and explanations for shapes like ellipses, hyperbolas, and parabolas. Understanding polar equations thus gives insight into the nature of these figures.

Conic Section

Conic sections represent an important class of curves in mathematics, derived from slicing a cone with a plane. The main types of conic sections include ellipses, parabolas, and hyperbolas. Each type has unique properties, which are best captured using both polar and rectangular coordinates.
The polar equation of a conic section defines its shape using a focus and directrix. Given in the form \( r = \frac{ed}{1 + e\cos\theta} \) or similar, where \( e \) represents eccentricity. If \( e < 1 \), it's an ellipse; if \( e = 1 \), a parabola; and \( e > 1 \), a hyperbola.
Understanding these characteristics helps visualize how these curves behave and connect with one another. Moreover, it offers insight on how to change the form of the equation from polar to rectangular and vice versa.

Conversion Formulas

When converting between polar and rectangular coordinates, conversion formulas are indispensable tools. The basic conversions are:

• For distance: \( r = \sqrt{x^2 + y^2} \)
• For angle: \( \theta = \tan^{-1}(\frac{y}{x}) \), provided \( x eq 0 \)
• Convert x: \( x = r\cos\theta \)
• Convert y: \( y = r\sin\theta \)

Using these, one can transform polar equations to rectangular forms, aiding in various applications like graphing and solving complex equations.
These formulas provide the bridge between the two systems. For example, substituting \( r \) and \( \cos \theta \) in the initial polar equation aids in arriving at a simplified rectangular equation. Though this transformation might first appear daunting, the familiar sight of x and y simplifies further manipulation and interpretation of the problem.

Simplification Steps

Simplifying the result after conversion is crucial for understanding the nature of the equation. First, substitute polar terms with their rectangular equivalents using conversion formulas. For instance, the substitution of \( r \) and \( \cos \theta \) into the conic's polar equation replaces complex fractions with simpler algebraic expressions.
After substitution, eliminating denominators is efficient by multiplying entire terms, streamlining the process. Once achieved, isolate terms containing roots or x and y. Further simplification often involves algebraic techniques like squaring, distributing, and combining like terms.
These steps can reveal the underlying structure of a conic section equation in rectangular form. In the end, obtaining the simplest form enables better comprehension and practical use, such as graphing or applying calculus techniques.