The general equation of a conic is
Guidance
Degenerate conic equations simply cannot be written in graphing form. There are three types of degenerate conics:

A singular point, which is of the form:
(x−h)2a+(y−k)2b=0 . You can think of a singular point as a circle or an ellipse with an infinitely small radius. 
A line, which has coefficients
A=B=C=0 in the general equation of a conic. The remaining portion of the equation isDx+Ey+F=0 , which is a line. 
A degenerate hyperbola, which is of the form:
(x−h)2a−(y−k)2b=0 . The result is two intersecting lines that make an “X” shape. The slopes of the intersecting lines forming the X are±ba . This is becauseb goes with they portion of the equation and is the rise, whilea goes with thex portion of the equation and is the run.
Example A
Transform the conic equation into standard form and sketch.
Solution: This is the line
Example B
Transform the conic equation into standard form and sketch.
Solution:
The point (2, 1) is the result of this degenerate conic.
Example C
Transform the conic equation into standard form and sketch.
Solution:
This is a degenerate hyperbola.
Concept Problem Revisited
When you intersect a plane with a two sided cone where the two cones touch, the intersection is a single 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 a line. 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 a degenerate hyperbola.
Vocabulary
A degenerate conic is 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
1. Create a conic that describes just the point (4, 7).
2. Transform the conic equation into standard form and sketch.
3. Can you tell just by looking at a conic in general form if it is a degenerate conic?
Answers:
1.
2.
3. 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 no
Practice
1. What are the three degenerate conics?
Change each equation into graphing form and state what type of conic or degenerate conic it is.
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Sketch each conic or degenerate conic.
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