Question

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{6}{2+\cos \theta}\)

Short answer

The eccentricity e of the conic is 1 and the distance p from the pole to the directrix is 3. The given equation represents an ellipse.

Step-by-step solution

Recognize the general form

Recognize that \(r = \frac{ep}{1±e \cos(\theta- \theta_0)}\) is the general form of the equation for a conic section in polar coordinates.

Compare the given equation to the general form

Compare the given equation \(r=\frac{6}{2+\cos \theta}\) to the general form. It can be seen that the denominator in the given equation is of the form 1 + ecos(θ). Therefore, there's a '+' sign in the formula, and correspondingly, e=1.

Calculate the distance from the pole to the directrix

The formula for p in the general equation is then \(\frac{6}{1+1}\)= 3. So, p= 3.

Identify the conic section & sketching the graph

Using these values in the general equation and comparing it with the given equation, we see that they match exactly. Thus, this equation represents a conic section that is an ellipse. By using these values of e and p, one can sketch the graph for this conic.

Key concepts

Eccentricity of Conic Sections

Eccentricity is a fundamental parameter that classifies conic sections into circles, ellipses, parabolas, and hyperbolas. It is a non-negative real number that describes how much a conic section deviates from being circular. The eccentricity, often denoted as 'e', can be intuitive: a circle has an eccentricity of 0 because it is perfectly round; an ellipse has an eccentricity between 0 and 1, indicating it is stretched into an oval shape; a parabola has an eccentricity of exactly 1, signifying its open arms extend infinitely; and a hyperbola has an eccentricity greater than 1, showing it has two separate curves or 'branches'.

In the exercise, the equation of the conic section is presented in polar coordinates as \( r=\frac{6}{2+\cos \theta} \). To determine the eccentricity from this equation, we compare it to the standard form \( r = \frac{ep}{1±e \cos(\theta - \theta_0)} \). For this specific exercise, the eccentricity is found to be 1. This suggests that the conic section is a parabola because parabolas have an eccentricity of exactly 1.

Understanding eccentricity is crucial because it helps us in identifying and distinguishing conic sections, which in turn is vital in graphing them and predicting their geometric properties.

Distance from Pole to Directrix

The distance from the pole (origin in polar coordinates) to the directrix of a conic section is another important characteristic. The directrix, a fixed straight line, serves as a reference for defining and drawing conics. For polar equations of conic sections, the distance is given by the variable 'p' in the equation \( r = \frac{ep}{1±e \cos(\theta - \theta_0)} \).

In our exercise, once the eccentricity, e, is determined to be 1, we then find the numeric value of 'p'. By analyzing the given equation \(r=\frac{6}{2+\cos \theta}\), we conclude that p equals 3. This is obtained by matching the equation to the standard form, realizing that the denominator (2+\cos \theta) equates to \(1+1\cos \theta\) where e=1. Thus, the distance from the pole to the directrix, p, is 3 units. It is particularly significant when sketching the conic, as it determines where the curve will lie in relation to the directrix.

Graphing Conic Sections

Graphing conic sections in polar coordinates involves plotting the set of points that satisfy the conic's equation. As discussed, our elliptical equation has an eccentricity of 1 and p equals 3, fitting the characteristics of a parabola. Knowing the eccentricity and distance to the directrix, we can proceed to graph the conic.

To sketch \( r=\frac{6}{2+\cos \theta} \), it's essential to consider several values of \( \theta \) and calculate the corresponding 'r' to plot the points. Since we're dealing with a parabola, we expect a symmetrical graph about the axis of symmetry, which, in this case, is the line \( \theta = 0 \) due to the cosine function. The vertex of the parabola is where it is closest to the pole, and 'p' indicates the focus's distance from the vertex. The directrix, lying perpendicular to the axis of symmetry, is behind the focus by distance 'p'.

These graphical elements help in manually plotting the conic section, although it is always helpful to use graphing utilities to confirm results. The process of graphing reinforces a student's understanding of the relationship between the algebraic equation of a conic and its geometric representation.