# Degenerate Conics

The general equation of a conic is $$\ A x^{2}+B x y+C y^{2}+D x+E y+F=0$$. This form is so general that it encompasses all regular lines, singular points and degenerate hyperbolas that look like an X.  This is because there are a few special cases of how a plane can intersect a two sided cone.  How are these degenerate shapes formed?

# Graphing Degenerate Conics

degenerate conic is a conic that does not have the usual properties of a conic. Degenerate conic equations simply cannot be written in graphing form.  There are three types of degenerate conics:

1. A singular point, which is of the form: $$\ \frac{(x-h)^{2}}{a}+\frac{(y-k)^{2}}{b}=0$$. You can think of a singular point as a circle or an ellipse with an infinitely small radius.
2. A line, which has coefficients $$\ A=B=C=0$$ in the general equation of a conic.  The remaining portion of the equation is $$\ D x+E y+F=0$$, which is a line.
3. A degenerate hyperbola, which is of the form: $$\ \frac{(x-h)^{2}}{a}-\frac{(y-k)^{2}}{b}=0$$. The result is two intersecting lines that make an “X” shape.  The slopes of the intersecting lines forming the X are $$\ \pm \frac{b}{a}$$. This is because $$\ b$$ goes with the $$\ y$$ portion of the equation and is the rise, while $$\ a$$ goes with the $$\ x$$ portion of the equation and is the run.

# Examples

Example 1

Earlier, you were asked how degenerate conics are formed.

Solution

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.

Example 2

Transform the conic equation into standard form and sketch.

$$\ 0 x^{2}+0 x y+0 y^{2}+2 x+4 y-6=0$$

Solution

This is the line $$\ y=-\frac{1}{2} x+\frac{3}{2}$$

Example 3

Transform the conic equation into standard form and sketch.

$$\ 3 x^{2}-12 x+4 y^{2}-8 y+16=0$$

Solution

\ \begin{aligned} 3 x^{2}-12 x+4 y^{2}-8 y+16&=0 \\ 3\left(x^{2}-4 x\right)+4\left(y^{2}-2 y\right) &=-16 \\ 3\left(x^{2}-4 x+4\right)+4\left(y^{2}-2 y+1\right) &=-16+12+4 \\ 3(x-2)^{2}+4(y-1)^{2} &=0 \\ \frac{(x-2)^{2}}{4}+\frac{(y-1)^{2}}{3} &=0 \end{aligned}

The point (2, 1) is the result of this degenerate conic.

Example 4

Transform the conic equation into standard form and sketch.

$$\ 16 x^{2}-96 x-9 y^{2}+18 y+135=0$$

Solution

\ \begin{aligned} 16 x^{2}-96 x-9 y^{2}+18 y+135=0 & \\ 16\left(x^{2}-6 x\right)-9\left(y^{2}-2 y\right) &=-135 \\ 16\left(x^{2}-6 x+9\right)-9\left(y^{2}-2 y+1\right) &=-135+144-9 \\ 16(x-3)^{2}-9(y-1)^{2} &=0 \\ \frac{(x-3)^{2}}{9}-\frac{(y-1)^{2}}{16} &=0 \end{aligned}

This is a degenerate hyperbola.

Example 5

1. Create a conic that describes just the point (4, 7).

Solution

$$\ (x-4)^{2}+(y-7)^{2}=0$$

# Review

1. What are the three degenerate conics?

Change each equation into graphing form and state what type of conic or degenerate conic it is.

1. $$\ x^{2}-6 x-9 y^{2}-54 y-72=0$$
2. $$\ 4 x^{2}+16 x-9 y^{2}+18 y-29=0$$
3. $$\ 9 x^{2}+36 x+4 y^{2}-24 y+72=0$$
4. $$\ 9 x^{2}+36 x+4 y^{2}-24 y+36=0$$
5. $$\ 0 x^{2}+5 x+0 y^{2}-2 y+1=0$$
6. $$\ x^{2}+4 x-y+8=0$$
7. $$\ x^{2}-2 x+y^{2}-6 y+6=0$$
8. $$\ x^{2}-2 x-4 y^{2}+24 y-35=0$$
9. $$\ x^{2}-2 x+4 y^{2}-24 y+33=0$$

Sketch each conic or degenerate conic.

1. $$\ \frac{(x+2)^{2}}{4}+\frac{(y-3)^{2}}{9}=0$$
2. $$\ \frac{(x-3)^{2}}{9}+\frac{(y+3)^{2}}{16}=1$$
3. $$\ \frac{(x+2)^{2}}{9}-\frac{(y-1)^{2}}{4}=1$$
4. $$\ \frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{4}=0$$
5. $$\ 3 x+4 y=12$$

To see the Review answers, open this PDF file and look for section 9.6.

# Vocabulary

Term Definition
Conic Conic sections are those curves that can be created by the intersection of a double cone and a plane. They include circles, ellipses, parabolas, and hyperbolas.
degenerate conic A degenerate conic is a conic that does not have the usual properties of a conic section. Since some of the coefficients of the general conic equation are zero, the basic shape of the conic is merely a point, a line or a pair of intersecting lines.
degenerate hyperbola A degenerate hyperbola is an example of a degenerate conic. Its equation takes the form $$\ \frac{(x-h)^{2}}{a}-\frac{(y-k)^{2}}{b}=0$$. It looks like two intersecting lines that make an “X” shape.