Distinguishing Conic Sections: General Form

We discussed in the last section how to distinguish conic sections from their graphing format. But sometimes they are not in their graphing format. You can still tell which conic section you're looking at by following this thought process:

Note: To use this process, you must have all the variables on the same side of the equation.

Question #1: Are both variables squared?

If they are not, you have a parabola
If they are, then ask question #2

Question #2: Do the squared terms have the same sign?

If they do not, you have a hyperbola
If they do, then ask question #3

Question #3: Do the squared terms have the same coefficient?

If they do, you have a circle
If they do not, you have an ellipse.

Examples:
1. x2+y2+ 6x - 2y = 5
All the variables are on the same side, so we can start asking our questions.
Question #1: Are both variables squared?
Yes, so it's not a parabola. Let's move on.
Question #2: Do the squared terms have the same sign?
Yes, they're both positive. It's not a hyperbola. Let's move on again.
Question #3: Do the squared terms have the same coefficient?
Yes, they are both an understood 1.
It's a circle.
2. 5x2+ y - 20x = -17
All the variables are on the same side, so we can start asking our questions.
Question #1: Are both variables squared?
No. The x is squared but not the y.
It is a parabola.
3. 9x2+18x - 50y - 241 = 25y2
Not all the variables are on the same side, so we need to move the 25y2 before we go through the questions:
9x2- 25y2+18x - 50y - 241 = 0
Question #1: Are both variables squared?
Yes, so it's not a parabola. Let's move on.
Question #2: Do the squared terms have the same sign?
No, 9x2 is positive and - 25y2 is negative.
It is a hyperbola.

Practice:Identify each conic section as a parabola, circle, ellipse, or hyperbola.
1) 9x2-18x+16y2+96y+9 = 0
2) -x2-4x+4y2+24y = -16
3) -y2+2x-8y = 22
4) x2-6x+y2+8y-3 = 0
5) -2x2+16x-2y2+8y = 6
Answers: 1) ellipse    2) hyperbola    3) parabola    4) circle    5) circle

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