The general equation of a conic is \begin{align*}Ax^2+Bxy+Cy^2+Dx+Ey+F=0\end{align*}
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: \begin{align*}\frac{(xh)^2}{a}+\frac{(yk)^2}{b}=0\end{align*}
(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 \begin{align*}A=B=C=0\end{align*}
A=B=C=0 in the general equation of a conic. The remaining portion of the equation is \begin{align*}Dx+Ey+F=0\end{align*}Dx+Ey+F=0 , which is a line. 
A degenerate hyperbola, which is of the form: \begin{align*}\frac{(xh)^2}{a}\frac{(yk)^2}{b}=0\end{align*}
(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 \begin{align*}\pm \frac{b}{a}\end{align*}±ba . This is because \begin{align*}b\end{align*}b goes with the \begin{align*}y\end{align*}y portion of the equation and is the rise, while \begin{align*}a\end{align*}a goes with the \begin{align*}x\end{align*}x portion of the equation and is the run.
Example A
Transform the conic equation into standard form and sketch.
\begin{align*}0x^2+0xy+0y^2+2x+4y6=0\end{align*}
Solution: This is the line \begin{align*}y=\frac{1}{2} x+\frac{3}{2}\end{align*}
Example B
Transform the conic equation into standard form and sketch.
\begin{align*}3x^212x+4y^28y+16=0\end{align*}
Solution: \begin{align*}3x^212x+4y^28y+16=0
\end{align*}
\begin{align*}3(x^24x)+4(y^22y)&=16\\ 3(x^24x+4)+4(y^22y+1)&=16+12+4\\ 3(x2)^2+4(y1)^2&=0\\ \frac{(x2)^2}{4}+\frac{(y1)^2}{3}&=0 \end{align*}
The point (2, 1) is the result of this degenerate conic.
Example C
Transform the conic equation into standard form and sketch.
\begin{align*}16x^296x9y^2+18y+135=0\end{align*}
Solution: \begin{align*}16x^296x9y^2+18y+135=0\end{align*}
\begin{align*}16(x^26x)9(y^22y)&=135\\ 16(x^26x+9)9(y^22y+1)&=135+1449\\ 16(x3)^29(y1)^2&=0\\ \frac{(x3)^2}{9}\frac{(y1)^2}{16}&=0\end{align*}
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.
\begin{align*}4x^2+8x+y^2+4y=0 \end{align*}
3. Can you tell just by looking at a conic in general form if it is a degenerate conic?
Answers:
1. \begin{align*}(x4)^2+(y7)^2=0\end{align*}
2.
\begin{align*}4x^2+8x+y^2+4y&=0\\ 4(x^22x)+(y^2+4y)&=0\\ 4(x^22x+1)+(y^2+4y+4)&=4+4\\ 4(x1)^2+(y+2)^2&=0\\ \frac{(x1)^2}{1}\frac{(y+2)^2}{4}&=0\end{align*}
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 \begin{align*}x^2\end{align*}
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.
2. \begin{align*}x^26x9y^254y72=0\end{align*}
3. \begin{align*}4x^2+16x9y^2+18y29=0\end{align*}
4. \begin{align*}9x^2+36x+4y^224y+72=0\end{align*}
5. \begin{align*}9x^2+36x+4y^224y+36=0\end{align*}
6. \begin{align*}0x^2+5x+0y^22y+1=0\end{align*}
7. \begin{align*}x^2+4xy+8=0\end{align*}
8. \begin{align*}x^22x+y^26y+6=0\end{align*}
9. \begin{align*}x^22x4y^2+24y35=0\end{align*}
10. \begin{align*}x^22x+4y^224y+33=0\end{align*}
Sketch each conic or degenerate conic.
11. \begin{align*}\frac{(x+2)^2}{4}+\frac{(y3)^2}{9}=0 \end{align*}
12. \begin{align*}\frac{(x3)^2}{9}+\frac{(y+3)^2}{16}=1 \end{align*}
13. \begin{align*}\frac{(x+2)^2}{9}\frac{(y1)^2}{4}=1 \end{align*}
14. \begin{align*}\frac{(x3)^2}{9}\frac{(y+3)^2}{4}=0 \end{align*}
15. \begin{align*}3x+4y=12\end{align*}