# 9.6 Degenerate Conics

- Page ID
- 1037

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 \(\mathrm{X}\). This is because there are a few special cases of how a plane can intersect a two sided cone. How are these degenerate shapeGraphing Degenerate Conicss formed?

## Graphing Degenerate Conics

A **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 \(\mathrm{X}\) 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

Earlier, you were asked how degenerate conics are formed. 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.

Transform the conic equation into standard form and sketch.

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

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

Transform the conic equation into standard form and sketch.

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

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

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

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

Transform the conic equation into standard form and sketch.

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

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

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

This is a degenerate hyperbola.

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

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

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. \(x^{2}-6 x-9 y^{2}-54 y-72=0\)

3. \(4 x^{2}+16 x-9 y^{2}+18 y-29=0\)

4. \(9 x^{2}+36 x+4 y^{2}-24 y+72=0\)

5. \(9 x^{2}+36 x+4 y^{2}-24 y+36=0\)

6. \(0 x^{2}+5 x+0 y^{2}-2 y+1=0\)

7. \(x^{2}+4 x-y+8=0\)

8. \(x^{2}-2 x+y^{2}-6 y+6=0\)

9. \(x^{2}-2 x-4 y^{2}+24 y-35=0\)

10. \(x^{2}-2 x+4 y^{2}-24 y+33=0\)

Sketch each conic or degenerate conic.

11. \(\frac{(x+2)^{2}}{4}+\frac{(y-3)^{2}}{9}=0\)

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

13. \(\frac{(x+2)^{2}}{9}-\frac{(y-1)^{2}}{4}=1\)

14. \(\frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{4}=0\)

15. \(3 x+4 y=12\)