Graphing Conic Sections

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:

Click to access conic_sections.pdf

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!

Solve the system of equations (conics)

Q:  Solve the system of equations (both of which are conic sections)
1:   x2 + y2 – 20x + 8y + 7 = 0
2:  9x2 + y2 + 4x + 8y + 7 = 0

A:  I am going to solve this by using the elimination method (since I see that I can cancel out all of the y’s)

I am going to multiply equation 1 by (-1) and then add it to equation 2:

2:  9x2 + y2 + 4x + 8y + 7 = 0

1: -x2 – y2 + 20x – 8y – 7 = 0   [after it has been multiplied by -1]

Now, add them together to get equation 3:

3:  8x2 +24x = 0

Now, equation 3 only has x’s so we can solve for x (by factoring):

(8x)(x + 3) = 0

So, x = 0 or -3

But, we aren’t done… Find the solution to this system of conics means that we are finding the point(s) of intersection — and it turns out there are two points of intersection since there were two solutions for x.  So, our points (solutions) are:

(0, ?)  and (-3, ?)

We need to find the y-coordinate of each solution separately.  We do this by plugging in the x value to either of the original equations (both will lead us to the same correct solution):

First x-value:  (0, ?)

x2 + y2 – 20x + 8y + 7 = 0

(0)2 + y2 – 20(0) + 8y + 7 = 0

y2  + 8y + 7 = 0

Solve for y by factoring:

(y + 7)(y + 1)=0

y = -7, -1

OH!  So this really means that there are more than two solutions…. When x = 0, we found two answers for y.

(0, -7) and (0, -1)

Second x-value:  (-3, ?)

x2 + y2 – 20x + 8y + 7 = 0

(-3)2 + y2 – 20(-3) + 8y + 7 = 0

9 + y2 +60 + 8y + 7 = 0

y2 + 8y + 76 = 0

Since this does not factor, we need to solve for y by using the quadratic formula.  I did not show my work here, but we end up with a negative under the square root, so there are no real solutions for when x = -3.

Therefore, there are two real solutions to this equation (2 points of intersection) and they are:

(0, -7) and (0, -1)

Here is a visual of what we are solving for….. We had the equations for the two graphed conic sections and we found the two points of intersection shown below: