1. 33. Conic Sections in General FormH E
33. Conic Sections in General Form P

2. General Form of a Conic Equation
We sometimes see conic equations written in General Form:
where A, B, C, D, E and F are integers and A, B and C are NOT ALL equal to zero.

3. A conic equation written in General Form doesn’t have to have all SIX terms! Several of the coefficients A, B, C, D, E and F can equal zero, as long as A, B and C don’t ALL equal zero.
If A, B and C all equal zero, what kind of equation do you have?
... T H I N
K...
Linear!

4. So, it’s a conic equation if...
the highest degree (power) of x and/or y is 2 (at least ONE has to be squared)
the other terms are either linear, constant, or the product of x and y
there are no variable terms with rational exponents (i.e. no radical expressions) or terms with negative exponents (i.e. no rational expressions)

5. What values form a circle? What values form an Ellipse?
The values of the coefficients in the conic equation determine the TYPE of conic.
What values form a circle? What values form an Ellipse?
What values form a Hyperbola?
What values form a Parabola?

6. Circles... where A & C have the SAME SIGN and they have the SAME VALUE
NOTE: There is no Bxy term, and D, E & F may equal zero!
For example:

7. Ellipses... where A & C have the SAME SIGN but DIFFERENT VALUES
NOTE: There is no Bxy term, and D, E & F may equal zero!
For example:

8. Hyperbola... where A & C have DIFFERENT signs. For example:
NOTE: There is no Bxy term, and D, E & F may equal zero!
For example:

9. Parabola... A Parabola can be oriented 2 different ways:
A parabola is vertical if the equation has an x squared term AND a linear y term; it may or may not have a linear x term & constant:
A parabola is horizontal if the equation has a y squared term AND a linear x term; it may or may not have a linear y term & constant:

10. Parabola …Vertical
The following equations all represent vertical parabolas in general form; they all have a squared x term and a linear y term:

11. Parabola …Horizontal
The following equations all represent horizontal parabolas in general form, they all have a squared y term and a linear x term:

12. Summary ... General Form of a Conic Equation:
where A, B, C, D, E and F are integers and A, B and C are NOT ALL equal to zero.
Identifying a Conic Equation:

13. Practice ... Identify each of the following equations as a(n):
(a) ellipse (b) circle (c) hyperbola
(d) parabola (e) not a conic
See if you can rewrite each equation into its Graphing Form!

14. Answers ...
(a) ellipse (b) circle (c) hyperbola (d) parabola (e) not a conic