Question

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. Please help

Answer 1

A) ellipse, horizontal directrix at a distance of
`(5)/(2)` units above the pole

Explanation:

I just took the test.

Answer 2

We have the following equation:


`r= (5)/(3+2sin(\theta))`

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:


`C(0, -2)`

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:


`6 \ units`

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 \ units`

4. Foci:

Let's find c as follows:


`c=\sqrt{a^(2)-b^(2)}=\sqrt{3^(2)-2^(2)}=√(5)`

Then the foci are:


`f_(1)=(0, √(5)-2)`


`f_(2)=(0, -√(5)-2)`