Q:  Identify the center, foci, vertices, and co vertices then graph the following conic section:
x² + 9y² = 36

A:  I am going to refer to the Conic Section guide that I had previously put together for students who I tutor:

This is a must have for those studying conic sections.  Click on it — it’s free 🙂

So, we look at our given equation:  x² + 9y² = 36 and recognize by its form that it is an ellipse.

To match the standard form of an ellipse, we need it equal, so divide both sides by 36 to get:

x²/36 + 9y²/36 = 36/36

Reduce to get:

x²/36 + y²/4 = 1

Refer to the Conic Section guide I linked to from above to follow along!

This is a horizontal ellipse (since the number under the x is larger than the number under the y)

The center is (0,0) since there are no values being subtracted from the x or the y.

The major axis:  a = sqrt(36) = 6  (in the x-direction)

The minor axis: b = sqrt(4) = 2  (in the y-direction)

The distance from the center to to foci can be found as follows:

c² = a² – b²

c² = 36 – 4

c² = 32

c = sqrt(32)

And, in simplest radical form, that is:

c = 4*sqrt(2)

So, the distance from the center to each foci (in the x-direction) is 4*sqrt(2)

Now, take all of the information from a, b and c found above to get the points:

Center: (0, 0)

Vertices:  (-6, 0) and (6, 0)

Co-Vertices:  (0, -2) and (0, 2)

Foci:  (-4*sqrt(2), 0) and (4*sqrt(2), 0)

Plot these points on a graph and then you’ve got it!