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. \(r=\frac{5}{5+3 \sin \theta}\)

Short answer

The eccentricity of the conic is \(-3/5\) and the distance from the pole to the directrix is \(-25/3\). It is a hyperbola with the branch below the directrix verified by a graphing utility.

Step-by-step solution

Identify the Conic

The given equation \(r=\frac{5}{5+3 \sin \theta}\) is in the form \(r=\frac{ep}{1+e \sin \theta}\). By comparing these two equations, we can see that \(e = -3/5\) and \(ep = 5\). Note that because \(e\) is negative, this is the equation of a hyperbola.

Find the Eccentricity and Distance to Directrix

Eccentricity \(e = -3/5\) and distance \(p = \frac{ep}{e} = \frac{5}{-3/5} = -25/3\). The minus sign shows the directrix is below the pole.

Graph the Conic

To graph, we take various values of \(\theta\) and find corresponding \(r\). For hyperbola, we consider only the \(r\) greater than 0. The graph will be a branch of hyperbola with pole as a focus. If \(r\) is less than 0, then it represents another branch on the opposite side of the directrix.

Verify using a graphing utility

Confirm your drawing using a graphing utility. Ensure the shape and positioning match with your finding.