Problem 32 - College Algebra

Question

For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. $$ r=\frac{2}{3+3 \sin \theta} $$

Step-by-Step Solution

Answer

The conic is a parabola with eccentricity 1.

Step 1: Identify the Conic Section

The given equation is in the polar form, \( r = \frac{2}{3 + 3 \sin \theta} \). Recognize that this is a conic section in polar coordinates. Compare it with the general form \( r = \frac{ed}{1 + e \sin \theta} \), where it represents different conics based on the eccentricity \( e \).

Step 2: Determine the Parameters

In the polar equation \( r = \frac{2}{3 + 3 \sin \theta} \), the denominator \( 3 + 3 \sin \theta \) suggests the comparison constant term \( 3 \) is linked with \( ed \), and the \( 3 \sin \theta \) part is related to \( e \). Thus, the eccentricity \( e = \frac{3}{3} = 1 \).

Step 3: Identify the Type of Conic

When the eccentricity \( e = 1 \), the conic is a parabola. Since \( e = 1 \), the equation \( r = \frac{2}{3 + 3 \sin \theta} \) represents a parabola.

Step 4: Convert to a Recognizable Parabola Form

Multiply both sides by \( 3 + 3\sin\theta \) to eliminate the denominator: \( r(3 + 3\sin\theta) = 2 \). This simplifies to \( 3r + 3r\sin\theta = 2 \). In Cartesian coordinates, \( r\sin\theta = y \), so \( 3r = 2 - 3y \).

Step 5: Simplify and Relate to Cartesian Coordinates

Express \( r \) in terms of \( x \) and \( y \): \( r = \sqrt{x^2 + y^2} \). Substitute into the equation: \( 3\sqrt{x^2 + y^2} + 3y = 2 \), leading to the Cartesian form after simplification.

Step 6: Identify Vertex, Focus, and Directrix

After converting and simplifying, identify significant features of the parabola. Standard form provides the vertex, and properties of the parabola give the focus and directrix. With \( e = 1 \), relate calculations to define these features in polar and Cartesian coordinates.

Key Concepts

Polar Coordinates

Polar coordinates are used to locate points on a plane using a distance and angle. Instead of finding a point based on horizontal and vertical distances (as in Cartesian coordinates), polar coordinates use:

• Distance from a reference point, known as the origin.
• An angle from a reference direction, typically the positive x-axis.

For example, in the equation given, \[ r = \frac{2}{3 + 3 \sin \theta} \]\( r \) represents the distance from the origin to any point on the conic section, and \( \theta \) is the angle from the x-axis. Polar coordinates are particularly useful for curves, such as circles and conics, that are fielded around a single point, making complex calculations simpler than using Cartesian coordinates.

Parabola

A parabola is a symmetrical curve that looks like a U or an inverted U. In the context of conic sections, a parabola is defined by its eccentricity. The given equation\[ r = \frac{2}{3 + 3 \sin \theta} \]is a parabolic form in polar coordinates when the eccentricity equals 1. Some essential features of a parabola include:

• Vertex: The highest or lowest point on the parabola.
• Focus: A point from which distances are measured to give the curve its shape.
• Directrix: A line opposite the parabola, used with the focus to define the curve.

When transferred from polar to Cartesian coordinates, finding these features helps in fully describing the parabola's structure on a graph.

Eccentricity

Eccentricity is a parameter associated with conic sections that helps determine their shape:

• If eccentricity \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), it is an ellipse.
• For \( e = 1 \), the conic section is a parabola.
• When \( e > 1 \), it becomes a hyperbola.

In the polar equation \[ r = \frac{2}{3 + 3 \sin \theta} \],the eccentricity calculated is \( e = 1 \),which confirms the conic section is a parabola. Eccentricity helps understand the curve's openness and shape, guiding how to plot or transform it into another form like Cartesian coordinates.

Cartesian Coordinates

Cartesian coordinates are used to locate points on a plane with horizontal and vertical axes, typically labeled as x and y. In many mathematical problems, converting between polar and Cartesian coordinates can simplify the process of graphing curves. This is done by transforming the polar equations into the Cartesian form using:

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

In our exercise, transforming \[ 3r + 3r\sin\theta = 2 \]to Cartesian coordinates involves the substitution \( r \sin \theta = y \), lending the equation to correspondingly simplify. The resulting Cartesian form allows clear identification of the parabola's vertex, focus, and directrix. This dual representation offers a complete understanding of the conic section's properties on different coordinate systems.