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. \(r=\frac{5}{-1+2 \cos \theta}\)

Short answer

The conic section represented by the equation is a hyperbola with an eccentricity of 2. The distance from the pole to the directrix is 5. This is confirmed by graphing the equation using a graphical utility.

Step-by-step solution

Identify the Conic Section

The given equation is \( r = \frac{5}{-1 + 2\cos\theta} \), rearrange this equation to the form: \( r = \frac{ep}{1 - e\cos\theta} \), where e represents the eccentricity, and p is the distance from the pole to the directrix. Comparing coefficients, we get e = 2 and p = -5.

Calculate Eccentricity

The eccentricity of the conic section is given by the value 'e'. From the previous step, we derived that e = 2, so the eccentricity of the conic section is 2.

Calculate Distance from Pole to Directrix

The distance from the pole to the directrix is given by the value '|p|'. From step 1, we derived that p = -5. Thus, the distance from the pole to the directrix is |-5| which is 5.

Sketch and Identify the Conic Section

A conic section with eccentricity e > 1 is a hyperbola. Therefore, the equation represents a hyperbola. To sketch the graph, use a graphical utility and insert the given equation. Since this requires the usage of a graphing utility, manual representation isn't feasible. Please follow the instructions for your specific graphing utility.

Verify the Results

By checking with a graphing utility, you should get a hyperbola with the given eccentricity and distance from the pole to the directrix.