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

Step-by-Step Solution

Answer

The conic is a polar parabola with vertex and focus at the origin and directrix at \( x = 5 \).

Step 1: Rearrange Polar Equation

The given equation is in polar form: \( r(1+\cos \theta)=5 \). Rearrange this equation to isolate \( r \), giving \( r = \frac{5}{1 + \cos \theta} \).

Step 2: Identify Conic Type

The equation \( r = \frac{5}{1 + \cos \theta} \) resembles the standard form of a conic section in polar coordinates, \( r = \frac{ed}{1 + e \cos \theta} \), which represents a conic with eccentricity \( e \). In this form, when \( e = 1 \), the conic is a parabola.

Step 3: Determine Characteristics of the Parabola

For a parabola in polar coordinates, when \( e = 1 \), the equation is \( r = \frac{d}{1 + \, \cos\, \theta} \). Since we have \( 1+\cos \theta \), this indicates the parabola opens to the left, towards \( \theta = \pi \). The directrix is \( x = 5 \), and the vertex is at the origin \((0,0)\). The focus is also at the origin due to the polar source.

Step 4: Graph the Parabola

To graph this parabola, plot the vertex at the origin. The focus is also at the origin. The directrix is a vertical line at \( x = 5 \). Draw a parabola opening towards \( \theta = \pi \) (to the left). Label the vertex, focus, and directrix appropriately.

Key Concepts

Polar Coordinates

Polar coordinates are a method of representing points on a plane using a distance and an angle. Imagine an invisible ray emanating from a central point, called the pole, similar to the origin in Cartesian coordinates. The distance of a point from this pole is termed as "radius," often denoted as \( r \).
The angle \( \theta \) measures the counter-clockwise rotation from a fixed direction, typically the positive x-axis, which represents our reference direction.

• In polar coordinates, a point \( P \) is represented as \( (r, \theta) \).
• The coordinates can describe almost any location using various values of \( r \) and \( \theta \).
• This system is particularly useful in conic sections like parabolas, ellipses, and hyperbolas.

Polar coordinates provide a natural way of describing sectors and circular paths, making them ideal for applications involving angles and directions.

Parabola

A parabola is a conic section created by slicing a cone parallel to one of its sides. It has a unique u-shaped curve that can open up, down, left, or right. In the context of polar coordinates, a parabola's orientation is determined by its equation.
The standard form of a polar equation for a parabola, when eccentricity \( e = 1 \), is \( r = \frac{ed}{1 + e \cos \theta} \). Here, the values of \( r \) and \( \theta \) define the parabola's shape and direction.

• The vertex of a parabola is its highest or lowest point depending on its orientation.
• In polar coordinates, this is commonly at the origin \((0,0)\).
• Parabolas have only one focus and one directrix.

This simplicity makes parabolas easy to identify and useful in practical applications like satellite dishes and parabolic microphones.

Graphing Conics

Graphing conic sections involves plotting curves that represent ellipses, parabolas, or hyperbolas on a coordinate plane. Each conic has unique characteristics that can be represented graphically.
For graphing parabolas using polar coordinates, follow these steps:

• First, identify the type of conic from its equation, such as a parabola when \( e = 1 \).
• Locate the vertex, directrix, and focus if applicable, which are markers of the curve's geometry.
• Plot the vertex and sketch the general curvature, ensuring it aligns with expected orientation.
• Use the directrix as a guide to ensure the shape opens correctly.

Proper graphing provides visual insights into the property of the conic, making their applications and calculations intuitive.

Eccentricity

Eccentricity is a numerical value that helps determine the shape of a conic section. It is denoted by \( e \), and its value establishes whether a conic is a circle, ellipse, parabola, or hyperbola.

• An eccentricity of \( e = 1 \) signifies a parabola, as in our polar equation example.
• For ellipses, \( 0 < e < 1 \), indicating their slightly oval shape.
• A value of \( e > 1 \) indicates a hyperbola, which has two distinct curves.

Determining eccentricity provides essential information on the conic's form, influencing how it is graphed and understood in mathematical analysis.

Directrix and Focus

In the study of conic sections, the directrix and focus are significant defining elements. The focus is a fixed point, while the directrix is a line associated with each conic.
For a parabola, these elements crucially define its shape.

• The directrix is a line perpendicular to the axis of symmetry of the parabola, acting as a boundary guiding its curve.
• The focus is a point that helps determine the curvature of the parabola. It lies along the axis of symmetry.
• In polar equations, the focus is commonly at the pole, influencing how the parabola opens and the directrix's position can further aid in representing it correctly on a plane.

Understanding these key components helps in accurately plotting and comprehending conic sections across varied applications, from physics simulations to architectural designs.