Describe the graph represented by the equation r=5/(3+2 sin theta). i know it’s an ellipse but i’m not sure if the there is a vertical or horizontal directrix.

We have the following equation:



If we graph this equation we realize that in fact this is an ellipse with major axis matching the y-axis. So we can recognize these characteristics:

1. Center of the ellipse: 

The midpoint C of the line segment joining the foci is called the center of the ellipse. So in this exercise this point is as follows:



2. Length of major axis:

The line through the foci is called the major axis, so in the figure if you go from -5, at the y-coordinate, and walk through this major axis to the coordinate 1, the distance you run is the length of the major axis, that is:



3. Length of minor axis:

The line perpendicular to the foci through the center is called the minor axis. So in the figure if you go from -2, at the x-coordinate, and walk through this minor axis to the coordinate 2, the distance you run is the length of the minor axis, that is:



4. Foci:

Let's find c as follows:



Then the foci are: