Problem 32 - College Algebra with Corequisite Support

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 vertex at (0,0), focus at (0,1/3), and directrix at y=-1/3.

Step 1: Identify the Conic Section

The given equation is in the form of polar coordinates: \( r = \frac{ed}{1 + e\sin\theta} \). From this, we can identify the type of conic section based on the coefficient of \( \sin \theta \). The equation \( r = \frac{2}{3 + 3\sin\theta} \) can be rewritten as \( \frac{2}{3(1 + \sin\theta)} \). This helps identify the conic section as a Limacon because of the polar nature not matching standard conics, but rewritten form indicates we investigate properties further.

Step 2: Simplify and Re-evaluate the Form

Rewriting the given equation to match standard polar conic sections forms \[ r = \frac{2/3}{1 + \sin \theta} \]. Comparing it to \( r = \frac{ed}{1 + e\sin\theta} \), we have \( ed = \frac{2}{3} \) and \( e = 1 \). If \( e = 1 \), it is a parabola. The orientation indicates the vertex is at the pole (origin in polar coordinates).

Step 3: Convert Key Features to Cartesian

Write the polar equation \( r = \frac{2}{3 + 3\sin\theta} \) in Cartesian coordinates. Since the direct comparative form is \( e = 1 \), this continues to represent a parabola. Converting gives insights into standard form features, but direct polar analysis here has shown properties sufficient for visual representation.

Step 4: Determine Vertex, Focus, and Directrix

Since the conic is a parabola, find its vertex, focus, and directrix. The vertex is at the pole, or \((0, 0)\) in polar coordinates. The focus is at \((0, \frac{1}{3})\), and the directrix is a line parallel to the x-axis at \( y = -\frac{1}{3} \). The direct parameters arise here by direct conic properties, since \( r = \frac{2/3}{1+\sin\theta} \) leads to focus via manipulation close space positions.

Step 5: Graph the Parabola

Plot the vertex at the pole, draw the parabola opening upwards reflecting the form given by the angle properties, and ensure the focus and directrix align properly with computed values. Use polar grid or sketch approximately on cartesian grid for plot emphasis.

Key Concepts

Polar Coordinates

Polar coordinates provide a way to describe the position of a point by using the distance from a reference point and an angle from a reference direction. This system is quite useful for problems involving circular or spiral patterns.

• The reference point is known as the pole, often represented by the origin in Cartesian coordinates.
• The angle is measured from a fixed ray, commonly the positive x-axis.

Polar coordinates are distinct from Cartesian coordinates, where a point is described using horizontal and vertical measurements. The form of an equation can sometimes help identify the type of conic section it represents, such as parabolas, ellipses, or hyperbolas. In the given exercise, the equation is in polar form, simplifying the process of graphing spherical or circular shapes.

Parabola

A parabola is a specific type of conic section, which can be easily identified by its U-shaped curve. In mathematics, it is often described in terms of its unique features like vertex and focus.

• Parabolas result from the intersection of a cone with a plane parallel to its side.
• They have a reflective property where light or signals can be focused to a single point, the focus.

Parabolas can open upwards, downwards, left, or right based on their equation. In the polar context of the exercise, the parabola is defined due to having the eccentricity \( e = 1 \), indicating its characteristic shape backed by symmetry around its axis.

Vertex and Focus

The vertex and focus are crucial elements when dealing with parabolas. The vertex is the tip of the parabola, indicating the point where it changes direction.

• For the problem at hand, the vertex is at the pole or origin \((0, 0)\) in polar coordinates, where the parabola begins.
• The focus is another significant point, positioned at \((0, \frac{1}{3})\) away from the vertex in this context.

This specific focus placement is an outcome of the geometric properties of the parabola, as it relates to the directrix, which in this problem is represented by a line parallel to the x-axis, at \( y = -\frac{1}{3} \). The directrix serves as a reference for how the parabola stretches from the vertex.

Cartesian Conversion

Converting polar equations to Cartesian form helps visualize and understand the geometric representation based on standard plotting methods. The transformation involves using the relationships between both coordinate systems:

• Convert the radial distance \( r \) and angle \( \theta \) into \( (x, y) \) coordinates.
• The formulae \( x = r \cos\theta \) and \( y = r \sin\theta \) are used in the conversion process.

This conversion makes it easier to sketch the conic section on a Cartesian grid, which relies on more familiar plotting techniques. In the exercise example, the transformation aids in aligning the focus and directrix directly into a Cartesian interpretation, giving a different perspective on the same conic section characterized by polar properties.