Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{4}{1+3 \sin \theta}\)

Step-by-Step Solution

Answer

The equation in rectangular form is \( x^2 + y^2 + 6y\sqrt{x^2 + y^2} + 9y^2 = 16 \).

Step 1: Identify the Conic Section

The given polar equation is of the form \( r = \frac{ed}{1 + e\sin\theta} \), which represents a conic section with the directrix parallel to the polar axis. Here, \( e = 3 \), indicating the conic is a hyperbola because \( e > 1 \).

Step 2: Substitute Polar to Rectangular Coordinates

Recall the relationships between polar and rectangular coordinates: \( x = r\cos\theta \) and \( y = r\sin\theta \), and \( r^2 = x^2 + y^2 \). Substitute \( r = \frac{4}{1 + 3\sin\theta} \) and \( \sin\theta = \frac{y}{r} \) into these equations.

Step 3: Clear the Denominator

Multiply both sides of the given equation by \( 1 + 3\sin\theta \) to clear the fraction: \( r(1 + 3\sin\theta) = 4 \).

Step 4: Substitute \( \sin\theta \) and Rearrange

Replace \( \sin\theta \) with \( \frac{y}{r} \) to get \( r + 3y = 4 \). Now substitute back \( r = \sqrt{x^2 + y^2} \): \( \sqrt{x^2 + y^2} + 3y = 4 \).

Step 5: Square Both Sides

To eliminate the square root, square both sides: \( (\sqrt{x^2 + y^2} + 3y)^2 = 4^2 \). Simplifying gives \( x^2 + y^2 + 6y\sqrt{x^2 + y^2} + 9y^2 = 16 \).

Step 6: Simplify the Equation

Further simplifications can be done by rearranging terms, but the key focus is to verify transformations match the desired geometry via simplification of coordinates; therefore calculator assistance or graphing could evaluate intersections.

Key Concepts

Conic Sections

A conic section can be described as the curve obtained by intersecting a cone with a plane. They are fundamental in understanding various geometric shapes and their properties. When the angle of the plane is varied, we get different types of curves, such as:

• A circle
• An ellipse
• A parabola
• A hyperbola

Each of these conic shapes results from specific characteristics and equations in both polar and rectangular coordinates. Understanding conic sections is crucial as they form the basis of many advanced concepts in mathematics.

Hyperbola

A hyperbola is one type of conic section formed when a plane cuts both halves of a double cone. It appears as two separate curves mirroring each other. In mathematical terms, it is defined when the eccentricity, denoted as \( e \), is greater than 1.

The standard form of a hyperbola in polar coordinates is given by \( r = \frac{ed}{1 + e \sin \theta} \) or similar variations, where \( e \) represents the eccentricity. If \( e > 1 \), as in our exercise, this confirms the conic as a hyperbola. Understanding these equations aids in converting these into rectangular forms, allowing deeper insight into geometric properties.

Polar Coordinates

Polar coordinates represent a way of describing locations on a plane using a distance from a reference point and an angle from a reference direction. It's different from the rectangular (or Cartesian) method that uses \( (x, y) \) coordinates.

In polar coordinates, each point is determined by \( (r, \theta) \), where:

• \( r \) is the radial distance from the origin.
• \( \theta \) is the angular coordinate, often measured in radians.

This system is particularly useful in scenarios involving circular and rotational symmetries, such as finding the shapes of conic sections like circles and hyperbolas. Understanding how to convert between polar and rectangular coordinates is critical in navigating between two-dimensional graphical representations.

Rectangular Coordinates

Rectangular coordinates, commonly known as Cartesian coordinates, use a grid system to locate points in a plane based on two axes, \( x \) and \( y \). Each point is denoted by an \( (x, y) \) pair.

This coordinate system is advantageous for displaying functions and equations representing straight lines and parabolic curves. Converting from polar to rectangular coordinates involves a few key transformations:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r^2 = x^2 + y^2 \)

These transformations are integral when working with polar equations of conic sections, allowing for a deeper understanding of the relationship between polar and rectangular representations.