Problem 59 - College Algebra

Question

Express each equation in polar form with \(r\) as a function of \(\theta\). $$ 16 x^{2}+24 x y+9 y^{2}=4 $$

Step-by-Step Solution

Answer

\[ r = \sqrt{\frac{4}{16 \cos^2 \theta + 24 \cos \theta \sin \theta + 9 \sin^2 \theta}} \]

Step 1: Identify the Rectangular Coordinates

The given equation \( 16x^2 + 24xy + 9y^2 = 4 \) is in terms of \(x\) and \(y\), which are rectangular coordinates. To convert it into polar form, we need to express it in terms of \(r\) and \(\theta\).

Step 2: Convert Rectangular Coordinates to Polar Coordinates

In polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\). Substitute these into the equation:\[ 16(r \cos \theta)^2 + 24(r \cos \theta)(r \sin \theta) + 9(r \sin \theta)^2 = 4 \]

Step 3: Simplify the Expression

Simplify each term in the equation:1. \(16(r \cos \theta)^2 = 16r^2 \cos^2 \theta\)2. \(24(r \cos \theta)(r \sin \theta) = 24r^2 \cos \theta \sin \theta\)3. \(9(r \sin \theta)^2 = 9r^2 \sin^2 \theta\)The equation becomes:\[ 16r^2 \cos^2 \theta + 24r^2 \cos \theta \sin \theta + 9r^2 \sin^2 \theta = 4 \]

Step 4: Factor Out \(r^2\)

Factor out \(r^2\) from the equation:\[ r^2 (16 \cos^2 \theta + 24 \cos \theta \sin \theta + 9 \sin^2 \theta) = 4 \]

Step 5: Solve for \(r^2\)

Divide both sides by the expression in the parentheses:\[ r^2 = \frac{4}{16 \cos^2 \theta + 24 \cos \theta \sin \theta + 9 \sin^2 \theta} \]

Step 6: Express \(r\) as a Function of \(\theta\)

Take the square root of both sides to solve for \(r\):\[ r = \sqrt{\frac{4}{16 \cos^2 \theta + 24 \cos \theta \sin \theta + 9 \sin^2 \theta}} \]

Step 7: Final Result

Thus, the equation in polar form with \(r\) as a function of \(\theta\) is:\[ r = \sqrt{\frac{4}{16 \cos^2 \theta + 24 \cos \theta \sin \theta + 9 \sin^2 \theta}} \]

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a common way to specify the location of points in a two-dimensional plane. In this system, each point is identified by two numbers: its horizontal distance (x-coordinate) from the origin along the x-axis and its vertical distance (y-coordinate) from the origin along the y-axis. This approach is practical because it aligns with the grid-like structure of many real-world applications, such as maps and graphs.
For example, the equation given in the exercise, \( 16x^2 + 24xy + 9y^2 = 4 \), uses x and y to describe the location and relationship of points. Understanding rectangular coordinates is crucial, as it sets the foundation for converting these coordinates into another system like polar coordinates, which can simplify the analysis of complex problems.

Conversion Between Coordinate Systems

Converting between coordinate systems, such as from rectangular to polar coordinates, involves translating the point's specification from one form to another. For rectangular coordinates \((x, y)\), the corresponding polar coordinates can be found using the equations:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

where \( r \) is the radial distance from the origin to the point, and \( \theta \) is the angle from the positive x-axis to the point.
In the context of the exercise, this conversion is essential. The rectangle equation \( 16x^2 + 24xy + 9y^2 = 4 \) was rewritten in polar form by substituting \( x = r \cos \theta \) and \( y = r \sin \theta \). This allows for easier manipulation with regards to angles and radius, which are often more intuitive in problems involving rotations and circular motion.

Algebraic Manipulation

Algebraic manipulation is a powerful technique used to simplify and solve mathematical equations. In the exercise, this method was applied extensively to convert and simplify the original rectangular equation into polar form. This process involves substituting known expressions, factoring, and simplifying terms.
After substituting \( x = r \cos \theta \) and \( y = r \sin \theta \) into the given equation, each term was expanded and then combined:

• \( 16(r \cos \theta)^2 \rightarrow 16r^2 \cos^2 \theta \)
• \( 24(r \cos \theta)(r \sin \theta) \rightarrow 24r^2 \cos \theta \sin \theta \)
• \( 9(r \sin \theta)^2 \rightarrow 9r^2 \sin^2 \theta \)

Finally, by factoring \( r^2 \), the equation was resolved to express \( r \) in terms of \( \theta \), culminating in the polar form solution \( r = \sqrt{\frac{4}{16 \cos^2 \theta + 24 \cos \theta \sin \theta + 9 \sin^2 \theta}} \). This technique not only simplifies the equation but also highlights relationships between the different terms.