Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r=\frac{8}{4-3 \quad \cos \theta}\)

Step-by-Step Solution

Answer

The conic is a hyperbola with eccentricity \( e = \frac{3}{4} \) and directrix \( d = \frac{32}{3} \).

Step 1: Identify the Conic Type

The general equation for the conic section in polar coordinates is given by \( r = \frac{ed}{1 - e \cos \theta} \), where \(e\) is the eccentricity and \(d\) is the directrix. Comparing this with the given equation \( r = \frac{8}{4 - 3 \cos \theta} \), we can rewrite it as \( r = \frac{8}{4(1 - \frac{3}{4} \cos \theta)} \). From this, \( e = \frac{3}{4} \) and the conic section is a hyperbola since \( e > 1 \).

Step 2: Determine the Eccentricity

The eccentricity \( e \) is the coefficient of \( \cos \theta \) in the standard form. From the previous step, we have \( e = \frac{3}{4} \).

Step 3: Calculate the Directrix

Given \( ed = 8 \) from comparing with \( r = \frac{ed}{1 - e \cos \theta} \), and using \( e = \frac{3}{4} \), solve for \( d \):\[d = \frac{8}{e} = \frac{8}{\frac{3}{4}} = \frac{8 \cdot 4}{3} = \frac{32}{3}.\] Hence, the directrix is \( \frac{32}{3} \).

Step 4: Identify the Focus

The focus of this conic section is located at the origin since this is a standard property of the conics in the form \( r = \frac{ed}{1 - e \cos \theta} \), without additional constants added to the right-hand side of the fraction.

Key Concepts

Polar Coordinates

Polar coordinates are a lovely alternative to the typical Cartesian system. Instead of describing the position of a point using fixed horizontal and vertical distances (like in the Cartesian system), polar coordinates use a distance from a fixed point (the origin) and an angle from a fixed direction. The combination of these two values allows you to pinpoint any location on a plane.

• The distance from the origin is known as the radial coordinate, usually denoted by \( r \).
• The angle is the angular coordinate, typically represented by \( \theta \).

One of the exciting things about polar coordinates is that they are perfectly suited for conic sections, which are often expressed as equations with \( r \) and \( \theta \). Conics can be circles, ellipses, parabolas, or hyperbolas, and polar coordinates can describe them all with elegance. This change in perspective can simplify both mathematical analysis and the visual understanding of these shapes.

Eccentricity

Eccentricity is an essential concept when working with conic sections. It's a measure of how much a conic section deviates from being a perfect circle. The value of eccentricity, denoted by \( e \), determines the type of conic you are dealing with.

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

In the exercise at hand, we calculated \( e = \frac{3}{4} \). Since \( e \) is less than 1, we are dealing with an ellipse. This understanding helps in sketching and analyzing the conic section accurately.
Eccentricity not only defines the shape but also provides insight into the directrix and how the conic section is positioned in space.

Directrix

The directrix is a fixed line used in the geometric definition of a conic section. It acts in tandem with the focus to provide rules of how distances play a fundamental role in defining each conic type. When polar equations are used, the directrix helps to specify exactly the location and orientation of the conic.

• In relation to the eccentricity \( e \), the directrix \( d \) is part of a formula \( r = \frac{ed}{1 - e \cos \theta} \).
• The value of \( d \) can be calculated using the equation \( ed = p \) where \( p \) is a constant derived from the conic's equation.

For the given problem, we found that the directrix is \( \frac{32}{3} \), a crucial part of describing the ellipse. This allows us to visualize and understand how the entire shape operates relative to both the focus and this special line.