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. \(r=\frac{-1}{1-\sin \theta}\)

Short answer

The given equation is of a hyperbola with an eccentricity of 1 and a distance from the pole to the directrix of -1.

Step-by-step solution

Identify the Conic

The equation \(r=\frac{-1}{1-\sin\theta}\) can be rewritten as \(r(1-\sin\theta)=-1\), or \(r-r\sin\theta=-1\). This takes the form of a conic section in polar coordinates, \(r=\frac{ep}{1-e\sin\theta}\), of a hyperbola with a focus at the pole.

Find the Eccentricity

This equation for a conic section in polar form contains \(e\), which stands for the eccentricity. By comparing the given equation, \(r=\frac{-1}{1-\sin\theta}\), with the standard form, it can be identified that \(e=1\).

Find the Distance from the Pole to the Directrix

The equation also contains \(p\), which represents the distance from the pole to the directrix. By comparing the given equation with the standard form, we can identify that \(p=-1\). Thus, the directrix of the hyperbola is one unit below the pole.

Sketch the Graph

A sketch of the hyperbola would reveal the eccentricity and directrix. The hyperbola is opened upwards, and the directrix is a horizontal line one unit below the pole.

Confirm with Graphing Utility

You can confirm your results by graphing \(r =\frac{-1}{1-\sin \theta}\) using a graphing utility. The graphical representation should confirm that this is indeed a hyperbola with a directrix one unit below the pole.