General Form of Conic Sections

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Textbook Reference: Any of sections 10.1 - 10.3 in our textbook.

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We've now studied all four main types of conic sections — circles, ellipses, parabolas, and hyperbolas — each of which has its own unique properties, and its own unique standard form equation.

\[ \begin{align*} &\text{Circle:} & (x-h)^2+(y-k)^2 &= r^2\\[2.5 ex] &\text{Ellipse:} & \dfrac{(x-h)^2}{r_x^2}+\dfrac{(y-k)^2}{r_y^2} &= 1\\[2.5 ex] &\text{Hyperbola:} & \dfrac{(x-h)^2}{r_x^2}-\dfrac{(y-k)^2}{r_y^2} &= 1 \text{ (horizontal)}\\ & & \dfrac{(y-k)^2}{r_y^2}-\dfrac{(x-h)^2}{r_x^2} &= 1 \text{ (vertical)}\\[2.5 ex] &\text{Parabola:} & x-h &= \dfrac{1}{4p}(y-k)^2 \text{ (horizontal)}\\ & & y-k &= \dfrac{1}{4p}(x-h)^2 \text{ (vertical)} \end{align*} \]

However, these shapes have just as many similarities as they have differences. This might seem surprising when talking about such seemingly disparate shapes, but it should actually make sense when you think about how they all come from the same basic idea: slicing through a double cone.

For example, notice that all of the above equations have one important thing in common: they're all quadratic in \(x\) and \(y\), that is, they have an \(x\) and/or \(y\) term being squared.

In fact, by expanding everything and collecting all the like terms, we can come up with a single equation that describes every single conic section: \[Ax^2+Bxy+Cy^2+Dx+Ey+F=0\] This is what we'll call the general form of a conic section, because any conic section can be expressed in this form.

For example, you should verify that the ellipse with equation \[\dfrac{(x-1)^2}{36}+\dfrac{(y+3)^2}{9}=1\] can be algebraically rearranged to give the general form \[4x^2+16y^2-8x+96y+4=0\text.\]

While it's generally pretty easy to go from standard form of a conic section to general form, going the other way can be a little more tricky. We'll look at how to do this in class, but we'll need to an algebraic technique you might not have seen come up in a while: completing the square.

If you need to review completing the square before our next class, I highly suggest this YouTube video that explains how it works in a very visual way!

Preview Activity 22

Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

1. Enter each of the following conic sections into Desmos, to find out whether it's a circle, ellipse, parabola, or hyperbola.
   1. \(2x^2+y^2+4x-6y-5=0\)
   2. \(x^2+6x+12y-15=0\)
   3. \(x^2+y^2-6x+8y+9=0\)
   4. \(x^2-y^2-6x+8y+9=0\)

2. Use what you learned from the previous question to complete the following flowchart.

   
3. Answer AT LEAST one of the following questions:
   1. What was something you found interesting about this activity?
   2. What was an "a-ha" moment you had while doing this activity?
   3. What was the muddiest point of this activity for you?
   4. What question(s) do you have about anything you've done?

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Posted by Bill Shillito