Problem 23 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recognize the Conic Section Formula

The given polar equation is \(r=\frac{3}{8-8 \cos \theta}\). We recognize this form as a conic section formula in polar coordinates, which is generally \(r = \frac{ed}{1 - e \cos \theta}\) or \(r = \frac{ed}{1 + e \cos \theta}\).

Step 2: Identify Eccentricity and Directrix

From comparing \(r = \frac{3}{8-8 \cos \theta}\) with \(r = \frac{ed}{1 - e \cos \theta}\), we identify: - \(ed = 3\)- \(1 - e = -8\) which translates to \(e = \frac{8}{8}\).Therefore, \(e = 1\) and \(d = \frac{3}{e} = 3\).

Step 3: Determine the Conic Type

We know that for a conic section:- If \(e < 1\), it's an ellipse.- If \(e = 1\), it's a parabola.- If \(e > 1\), it's a hyperbola.Since \(e = 1\), the conic is a parabola.

Step 4: Identify the Directrix

In the equation \(r = \frac{ed}{1 - e\cos\theta}\), the directrix is given by the line \(x = -d\) (because the focus is at the origin and \(e = 1\) in this case). Thus, the directrix is \(x = -3\).

Key Concepts

Conic Sections

Conic sections are curves that are formed by the intersection of a plane and a double napped cone. This geometric figure can manifest in different shapes, depending on the angle and position of the intersecting plane relative to the cone.
Conic sections include:

• Circle: When the plane cuts through perpendicular to the axis of the cone.
• Ellipse: When the plane cuts through at an angle, but does not touch the base.
• Parabola: When the plane is parallel to one of the generating lines of the cone.
• Hyperbola: When the plane cuts through both halves of the double cone.

Each type of conic section has unique properties and equations associated with it. Understanding conic sections in polar coordinates can help in visualizing and analyzing the curves based on their algebraic equations, like the one in the original exercise.

Eccentricity

Eccentricity (\(e\)) is a number that describes how "stretched" or "circular" a conic section is. It determines the conic's shape and can be defined as the ratio of the distance between the focus and any point on the conic to the perpendicular distance from the point to the directrix.
For different conic sections, eccentricity values differ:

• Circle: \(e = 0\)
• Ellipse: \(0 < e < 1\)
• Parabola: \(e = 1\)
• Hyperbola: \(e > 1\)

In the exercise, since \(e = 1\), it indicates a parabolic shape.
Understanding eccentricity can help in distinguishing between the types of conics, predicting their shapes, and simplifying complex geometric problems.

Directrix

The directrix is a fixed line used in the geometric description of a curve, especially in defining conics.
In conic sections, the directrix helps in defining the shape and position of the conic relative to its focus. For a given point on a conic section, the ratio of the distance to the focus and the distance to the directrix remains constant, equaling the eccentricity (\(e\)) of the conic.

• For a parabola, this takes the simplest form: the distance from any point on the parabola to its focus is equal to the distance from the point to the directrix, since \(e = 1\).
• In the given exercise, the directrix is the line \(x = -3\)

This makes understanding directrix crucial for identifying and working with conic sections in polar coordinates.