Problem 21 - College Algebra

Question

For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{3}{2+5 \cos \theta} $$

Step-by-Step Solution

Answer

The rectangular equation corresponding to the polar conic is derived through substitution and manipulation but is complex due to having the focus at the pole.

Step 1: Recognize the Polar Equation

The given polar equation is \( r = \frac{3}{2 + 5 \cos \theta} \). This is the standard form of a conic section with the focus at the pole. We need to convert this to rectangular coordinates \((x, y)\).

Step 2: Understand Polar to Rectangular Conversions

Remember from polar to rectangular transformations: \( x = r \cos \theta \) and \( y = r \sin \theta \). Also, \( r^2 = x^2 + y^2 \) and \( \cos \theta = \frac{x}{r} \).

Step 3: Replace \( r \) and \( \cos\theta \) in the Polar Equation

Substitute \( r = \sqrt{x^2 + y^2} \) and \( \cos \theta = \frac{x}{r} = \frac{x}{\sqrt{x^2+y^2}} \) into the polar equation: \( \sqrt{x^2+y^2} = \frac{3}{2 + 5 \frac{x}{\sqrt{x^2+y^2}}} \).

Step 4: Simplify and Solve for \( r \)

Multiply both sides by the denominator \(2 + 5 \frac{x}{\sqrt{x^2 + y^2}}\) to isolate \(r\): \( \sqrt{x^2+y^2}(2 \sqrt{x^2 + y^2} + 5x) = 3 \).

Step 5: Square Both Sides

Square both sides to eliminate the square root: \( x^2 + y^2(2 \sqrt{x^2+y^2} + 5x)^2 = 9 \).

Step 6: Solve for Rectangular Equation

Separate terms and simplify the equation to express it in \(x\) and \(y\) without the variable \(r\), which will result in a rectangular equation that represents the same conic section as the original polar equation.

Key Concepts

Polar Coordinates

Polar coordinates are a system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. This system uses two numbers:

• The radial coordinate, often denoted as \( r \), which represents the distance from the origin to the point.
• The angular coordinate, denoted as \( \theta \), indicating the angle between the positive x-axis and the line connecting the origin to the point.

This provides a unique way of locating points which can be especially handy when dealing with curves and circular shapes, like conic sections. Polar coordinates are particularly useful in situations where symmetry around a point is involved, such as the orbits of planets or the behavior of waves. By understanding polar equations, one can easily work with circles, spirals, and other shapes that are naturally circular or radial. In this exercise, we start with a polar equation of a conic section where \( r = \frac{3}{2 + 5 \cos \theta} \), indicating a relationship that is dependent on the angle \( \theta \) and can be visualized as a curve originating from the pole, or origin.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two values to determine a point's location on a plane. These are:

• The x-coordinate, which indicates the horizontal position.
• The y-coordinate, which indicates the vertical position.

With rectangular coordinates, every point is defined by a perpendicular intersection of its x and y values. Additionally, the relationship between x and y often comes in the form of algebraic equations that are easy to graph and interpret.
In this exercise, converting the polar equation to rectangular coordinates involves converting \( r \) and \( \cos \theta \) to their corresponding x and y expressions. Using the Pythagorean identity \( r^2 = x^2 + y^2 \), we can substitute \( r \) and \( \theta \) values into the polar equation to express it in terms of x and y. This provides a different perspective on the conic section as it can now be visualized on a standard Cartesian plane, allowing for easier manipulation and understanding of its shape and behavior.

Coordinate Transformation

Coordinate transformation is the process of converting coordinates from one system to another. In this scenario, we transition from polar to rectangular coordinates. This transformation is crucial because some equations or problems are easier to solve or understand in one coordinate system compared to another.

• The key relationships are: \( x = r \cos \theta \) and \( y = r \sin \theta \), which translate polar coordinates into rectangular ones.
• Additionally, \( r \) can be expressed in terms of x and y by \( r = \sqrt{x^2 + y^2} \).
• And the angle \( \theta \) is given by \( \cos \theta = \frac{x}{r} \), translating angular relationships into Cartesian form.

By performing these substitutions, the exercise involves systematically rewriting the polar coordinates in terms of their equivalent rectangular forms. This technical ability allows students to discern more about the properties of the conic sections by observing it in a more familiar coordinate system like the rectangular one. Transformations make it easier to analyze and graph these functions in ways that provide more intuitive insights about their nature.