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Conics: Classifying from General Equation

A conic section is the cross section of a plane and a double napped cone. The four basic conic sections do not pass through the vertex of the cone. These are:



Circle - the intersection of the cone and a perpendicular plane.

Ellipse - the intersection of the cone and a plane that is neither perpendicular nor parallel and cuts through the width of the cone.

Parabola - the intersection of the cone and a plane that is neither perpendicular nor parallel and cuts through the top and a side of the cone.

Hyperbola - the intersection of a cone and a plane that is parallel to the cone and cuts through the top and bottom of the cone.

Each conic section has a unique standard equation but all of them can be described by the general equation:

GENERAL EQUATION OF A CONIC:


A x 2 +B y 2 +Cx+Ey+F=0


where A, B, C, D and E are constants.



The values of the constants A and C in the general equation reveal the type of conic the equation is describing.

CLASSIFYING A CONIC FROM GENERAL EQUATIONS:


1. Circle:          A = C

2. Parabola:     AC = 0      where A = 0 or C = 0 but not both

3. Ellipse:         AC > 0     where A and C have like signs

4. Hyperbola:   AC < 0     where A and C have unlike signs

This test only applies if the graph of the equation is a conic. It does not apply to equations that have no real graph, such as x2 + y2 = -1.


Let's use these classification rules in some examples:

Example 1: Classifying the graph of each equation.
a) 5 x 2 −8x+y−10=0      b) 6 x 2 −2 y 2 +9x−3y+4=0

Step 1: Classify equation 5 x 2 −8x+y−10=0 by determining the values of A and C and applying the general equation rule.


Because the equation has no y2-term, c = 0.

5 x 2 −8x+y−10=0


A = 5, C = 0


AC=5·0=0


Apply Rule:


Because AC = 0 and both A and C are not zero, Rule 2 applies. The graph is a parabola.

Step 2: Classify equation 6 x 2 −2 y 2 +9x−3y+4=0 by determining the values of A and C and applying the general equation rule.

6 x 2 −2 y 2 +9x−3y+4=0


A = 6, C = -2


AC=6·−2=−12


Apply Rule:


Because AC = -12 < 0 and A and C have unlike signs, Rule 4 applies. The graph is a hyperbola.

Example 2: Classifying the graph of each equation.
a) 3 x 2 +3 y 2 −5x+2y=0      b) 4 x 2 +2 y 2 −6y+14=0

Step 1: Classify equation 3 x 2 +3 y 2 −5x+2y=0 by determining the values of A and C and applying the general equation rule.

3 x 2 +3 y 2 −5x+2y=0


A = 3, C = 3


A = C


Apply Rule:


Because A = C, Rule 1 applies. The graph is a circle.

Step 2: Classify equation 4 x 2 +2 y 2 −6y+14=0 by determining the values of A and C and applying the general equation rule.

4 x 2 +2 y 2 −6y+14=0


A = 4, C = 2


AC=4·2=8


Apply Rule:


Because AC = 8 > 0 and A and C have like signs, Rule 2 applies. The graph is an ellipse.





Related Links:
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