Problem 11 - College Algebra

Question

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{3}{10+10 \cos \theta} $$

Step-by-Step Solution

Answer

The conic is a parabola with directrix \( x = 3 \) and eccentricity \( e = 1 \).

Step 1 - Identify the type of conic section

The given polar equation is of the form \( r = \frac{ed}{1 + e \cos \theta} \), which corresponds to a conic section with the focus at the origin. This form indicates either an ellipse, a parabola, or a hyperbola, depending on the value of \( e \) (the eccentricity).

Step 2 - Compare given equation to standard form

The given equation is \( r = \frac{3}{10 + 10 \cos \theta} \). Upon comparison with the standard form, we have \( ed = 3 \) and \( e \times 10 = 10 \).

Step 3 - Solve for eccentricity (e)

From \( e \times 10 = 10 \), simplify to find \( e = 1 \). Since the eccentricity \( e = 1 \), this conic is a parabola.

Step 4 - Determine the directrix

Since \( ed = 3 \) and \( e = 1 \), apply \( ed = d \), therefore \( d = 3 \). The directrix is a line perpendicular to the axis of symmetry at a distance \( d = 3 \) units from the focus.

Key Concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system that represent points in the plane using an angle and a distance from a fixed point, usually called the pole or the origin. Instead of describing locations with a grid system like Cartesian coordinates, polar coordinates allow us to specify a point's position with two parameters:

• The radial distance, \( r \), which is the distance from the pole to the point.
• The angle, \( \theta \), which is the counterclockwise angle from the positive x-axis to the radial line connecting the pole to the point.

Using polar coordinates is particularly useful in problems involving circular motion or symmetry about a point, such as dealing with conic sections. Conic sections can have their equations expressed in polar form, which simplifies understanding and categorizing these curves.

Eccentricity

Eccentricity is a measure that describes the shape of a conic section, such as a circle, ellipse, hyperbola, or parabola. It is denoted by \( e \). This important parameter helps in determining the type of conic section as follows:

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

In the context of conics described by a polar equation \( r = \frac{ed}{1 + e \cos \theta} \), the value of \( e \) indicates the conic's shape. A key task in studying conic sections is finding this eccentricity, as it reveals crucial information about the conic's properties.

Parabola

A parabola is a conic section that can be described as the set of all points equidistant from a single point, called the focus, and a line, known as the directrix. In terms of eccentricity, a parabola is unique because:

• The eccentricity \( e = 1 \), tightly defining its characteristic bow-like shape.
• It is symmetric around a principal axis called the axis of symmetry.
• Unlike ellipses and hyperbolas, parabolas are infinite curves, extending indefinitely outward in both directions.

Parabolas arise naturally in many real-world scenarios, such as the paths of projectiles under gravity. In polar form, a parabola is represented by an equation where the eccentricity is exactly 1, making it straightforward to identify.

Directrix in Conics

The directrix in conics is a reference line unique to each conic section, except circles. It is important for defining the locus of points forming the conic. Here's what we should know about the directrix:

• For parabolas, it is the line from which distances to any point on the parabola are equal to the distances from the parabola's focus.
• In the equation \( r = \frac{ed}{1 + e \cos \theta} \), \( d \) represents the directrix's distance from the focus along the axis of the parabola.
• The directrix helps in determining the orientation and positioning of the conic in both Cartesian and polar coordinates.

Understanding the directrix is crucial, especially when analyzing and constructing conics based on given parameters. It serves as an invisible guideline, shaping the conic's structure and position in space.