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!