# Distinguishing Conic Sections: General Form

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

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

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.

If they do not, you have an ellipse.

**Examples:**

1. x

^{2}+y^{2}+ 6x - 2y = 5
All the variables are on the same side, so we can start asking our questions.

Question #1: Are both variables squared?

2. 5xQuestion #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.

It's a circle.

^{2}+ y - 20x = -17
All the variables are on the same side, so we can start asking our questions.

Question #1: Are both variables squared?

3. 9xQuestion #1: Are both variables squared?

No. The x is squared but not the y.

It is a parabola.

It is a parabola.

^{2}+18x - 50y - 241 = 25y^{2}
Not all the variables are on the same side, so we need to move the 25y

9x

Question #1: Are both variables squared?

^{2}before we go through the questions:9x

^{2}- 25y^{2}+18x - 50y - 241 = 0Question #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, 9x

It is a hyperbola.

^{2}is positive and - 25y^{2}is negative.It is a hyperbola.

**Practice:**Identify each conic section as a parabola, circle, ellipse, or hyperbola.
1) 9x

2) -x

3) -y

4) x

5) -2x

^{2}-18x+16y^{2}+96y+9 = 02) -x

^{2}-4x+4y^{2}+24y = -163) -y

^{2}+2x-8y = 224) x

^{2}-6x+y^{2}+8y-3 = 05) -2x

^{2}+16x-2y^{2}+8y = 6**Answers:**1) ellipse 2) hyperbola 3) parabola 4) circle 5) circle

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