Question

Describe the graph represented by the equation r=5/(3+2 sin theta).

Answer 1

We have the following equation:

$r= \frac{5}{3+2sin(\theta)}$

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:

$c=\sqrt{a^{2}-b^{2}}=\sqrt{3^{2}-2^{2}}=\sqrt{5}$

Then the foci are:

$f_{1}=(0, \sqrt{5}-2)$

$f_{2}=(0, -\sqrt{5}-2)$