Problem 39 - 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. \(r(1+\cos \theta)=5\)

Step-by-Step Solution

Answer

The conic section is a parabola with vertex \((2.5, 0)\), focus \((5, 0)\), and directrix \(x = 0\).

Step 1: Identify the Conic Section

The given polar equation is of the form \(r(1+e\,\cos \theta) = d\), where \(d = 5\) and \(e\) represents the eccentricity. Comparing to the standard form, \(e = 1\). Since \(e = 1\), this conic represents a parabola.

Step 2: Convert Polar Equation to Cartesian

To find more details about the parabola, we convert it to a Cartesian form. Start by expanding and rewriting the equation: \(r = \frac{5}{1 + \cos \theta}\). We know that \(\cos \theta = \frac{x}{r}\), so substituting gives \(r = \frac{5r}{r+x}\). Simplifying, we obtain \(r = 5 - x\), hence, \( r - x = 5\). Convert to Cartesian: \(\sqrt{x^2 + y^2} = 5 - x\); squaring both sides, we have the equation \((x - 2.5)^2 + y^2 = 2.5^2\).

Step 3: Graph the Parabola

The equation \((x - 2.5)^2 + y^2 = 2.5^2\) describes a circle. However, since we recognize the polar form, the parabola opens to the right. Its vertex is \((2.5, 0)\). Calculations for focus: Distance from vertex to focus is half the directrix-to-vertex length, which is 5, focusing at \((5, 0)\). Directrix is at \(x = 0\).

Step 4: Label the Graph

On the graph, plot the vertex at \((2.5, 0)\), focus at \((5,0)\), and draw a line for the directrix at \(x = 0\). Ensuring the parabola is drawn opening to the right, where the distance from any point on the parabola to the focus (\((5,0)\)) equals the distance to the line \(x = 0\).

Key Concepts

Parabola

A parabola is a unique type of conic section characterized by its U-shape. It can open in a direction like up or down, or even sideways. A parabola comprises several key components:
* **Vertex**: This is the point where the parabola changes direction. For our exercise, the vertex found is at \((2.5, 0)\).
* **Focus**: This is a point from which distances to the parabola are measured. In our solved problem, the focus is at \((5, 0)\).
* **Directrix**: A fixed line that helps define the parabola, in our case, it is at \(x = 0\).

All points on a parabola are equidistant from the focus and directrix, giving it the distinct shape we recognize. Understanding these attributes helps us interpret and graph it accurately.

Polar Coordinates

Polar coordinates offer a different way of describing points on a plane, focusing on distance and angle. These are quite useful when dealing with curves, such as the conic section depicted by the formula \(r(1+\cos \theta)=5\).

In polar coordinates:
* **\(r\)** represents the radial distance from a central point (origin).
* **\(\theta\)** is the angle, measured from the positive x-axis.

This coordinate system is especially helpful in problems involving curves centered around a point, where traditional grid-like Cartesian coordinates may not be as intuitive. This helps to initially identify our type of conic section—here, a parabola.

Cartesian Coordinates

Cartesian coordinates are the most common way to represent points on a plane using two perpendicular axes: typically, x and y. In our problem, once we've identified the conic section type using polar coordinates, we shift to Cartesian.

Key elements of Cartesian coordinates include:
* **X-axis** and **Y-axis**, which define the plane.
* Any point's location given as \((x, y)\).

This method is beneficial when graphing, as it translates geometric features more naturally. For our problem, converting from the equation \(r = 5 - x\) to \((x - 2.5)^2 + y^2 = 2.5^2\) gives us a standard form to graph and locate specific parabola features.

Graphing Conic Sections

Graphing conic sections, whether a circle, ellipse, parabola, or hyperbola, involves understanding each shape's unique properties and translating those onto a grid.

For graphing a parabola, remember:
* Begin with pinpointing the vertex \((2.5, 0)\) in this scenario, as the starting point of the graph.
* Mark the focus and the directrix to shape the curve.The process:

• From the vertex, guide the curve's direction toward the focus.
• Keep all points equidistant from the focus and the directrix.

Being comfortable in shifting between coordinate systems—polar for identification and Cartesian for drawing—enhances precision in graphing and understanding their nuances.