9.6. Degenerate Conics http://www.ck12.org
Concept Problem Revisited
When you intersect a plane with a two sided cone where the two cones touch, the intersection is asingle point. When
you intersect a plane with a two sided cone so that the plane touches the edge of one cone, passes through the central
point and continues touching the edge of the other conic, this produces aline. When you intersect a plane with a two
sided cone so that the plane passes vertically through the central point of the two cones, it produces adegenerate
hyperbola.
Vocabulary
Adegenerate conicis a conic that does not have the usual properties of a conic. Since some of the coefficients of
the general equation are zero, the basic shape of the conic is merely a point, a line or a pair of lines. The connotation
of the word degenerate means that the new graph is less complex than the rest of conics.
Guided Practice
- Create a conic that describes just the point (4, 7).
- Transform the conic equation into standard form and sketch.
− 4 x^2 + 8 x+y^2 + 4 y= 0 - Can you tell just by looking at a conic in general form if it is a degenerate conic?
Answers:
1.(x− 4 )^2 +(y− 7 )^2 = 0
− 4 x^2 + 8 x+y^2 + 4 y= 0
− 4 (x^2 − 2 x)+(y^2 + 4 y) = 0
− 4 (x^2 − 2 x+ 1 )+(y^2 + 4 y+ 4 ) =− 4 + 4
− 4 (x− 1 )^2 +(y+ 2 )^2 = 0
(x− 1 )^2
1 −(y+ 2 )^2
4 =^0- In general you cannot tell if a conic is degenerate from the general form of the equation. You can tell that
the degenerate conic is a line if there are nox^2 ory^2 terms, but other than that you must always try to put the conic
equation into graphing form and see whether it equals zero because that is the best way to identify degenerate conics.