Problem 21 - College Algebra

Question

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 for the given polar conic is \( x^2 + y^2 = 3 \).

Step 1: Identify Parameters

In the polar equation \( r = \frac{3}{2 + 5 \cos \theta} \), identify \( e \) (eccentricity) and \( p \) (semi-latus rectum). Here, \( e = 5 \) and \( ep = 3 \), so \( p = \frac{3}{5} \).

Step 2: Substitute Polar Coordinates

For conversion to rectangular form, use substitutions: \( r \cos \theta = x \) and \( r \sin \theta = y \). Express \( r \) as \( r = \sqrt{x^2 + y^2} \) and \( \cos \theta = \frac{x}{r} \).

Step 3: Express in Terms of \( x \) and \( y \)

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

Step 4: Eliminate Square Root

Multiply both sides by \( \sqrt{x^2 + y^2}(2 + \frac{5x}{\sqrt{x^2 + y^2}}) \) to eliminate the denominator and square root, leading to: \( \sqrt{x^2 + y^2} \cdot \sqrt{x^2 + y^2} = 3 \), or \( x^2 + y^2 = 3 \).

Step 5: Complete and Simplify Equation

Substitute back the equation formed, which becomes: \( 3rac{x}{\sqrt{x^2 + y^2}} + 2\sqrt{x^2 + y^2} = 3 \), simplify by multiplying the whole equation by \( \sqrt{x^2 + y^2} \) to get \( 5x + 2\sqrt{x^2 + y^2}^2 = 3\sqrt{x^2 + y^2} \), and eventually solve for x and y.

Key Concepts

Conic Sections

Conic sections are shapes created by slicing a cone with a plane at different angles. The main types of conic sections are circles, ellipses, parabolas, and hyperbolas. Each shape has unique characteristics that define its structure and properties. The equation given in the problem represents a conic section in polar form.

• **Circle** - Formed when the plane cuts perpendicular to the cone's axis. Equidistant from the center.
• **Ellipse** - An elongated circle; further from the center, but not at infinity.
• **Parabola** - Formed when the plane is parallel to the cone's side.
• **Hyperbola** - When the plane cuts both halves of the cone.

In polar coordinates, these are represented with equations involving the radial distance and angle. Understanding conic sections helps contextualize geometry and real-world problems.

Polar Coordinates

Polar coordinates use a combination of distance from a fixed point and angle from a fixed direction to locate points in space. Unlike the rectangular (or Cartesian) coordinates which are (x, y), polar coordinates are written as (r, \(\theta\)). Here:

• '\( r \)' is the distance from the pole (origin)
• '\( \theta \)' is the angle from the positive x-axis

This system is particularly useful in fields requiring circular symmetry, like physics and engineering. The exercise involves transforming a polar coordinate-based equation into a more conventional rectangular form, where the focus shifts on x and y coordinates. Converting between these systems allows for different mathematical approaches and solutions.

Rectangular Equation

A rectangular equation is expressed in terms of x and y coordinates, offering a straightforward application in algebra and calculus. This is the form most students are familiar with. In the given problem, we start with a polar equation and convert it to the rectangular equation format. This involves algebraic manipulation and understanding the relationship between polar and rectangular systems.

To convert:

• Utilize the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \).
• Replace \( r \) with \( \sqrt{x^2 + y^2} \) in the polar equation.

These steps transition between coordinate systems, removing dependence on angular components. The result is an equation in terms of x and y, familiar ground for solving and graphing in mathematics.

Eccentricity

Eccentricity is a parameter that determines the shape of a conic section. It indicates how much a conic section deviates from being circular. Each conic type has a specific range for its eccentricity value:

• **Circle**: Eccentricity (\(e\)) is 0.
• **Ellipse**: Eccentricity is greater than 0 but less than 1.
• **Parabola**: Eccentricity is exactly 1.
• **Hyperbola**: Eccentricity is greater than 1.

In our exercise, the eccentricity \(e = 5 \) signifies a hyperbola. Understanding eccentricity helps in identifying and distinguishing between different conic sections based on how elongated or flat they are. This concept is crucial when analyzing or graphing these curves in various coordinate systems.