# 9.6 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 $$\mathrm{X}$$. 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

##### Example 1

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.

##### Example 2

Transform the conic equation into standard form and sketch.

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

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

##### Example 3

Transform the conic equation into standard form and sketch.

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

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

\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.

##### Example 4

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.

##### Example 5

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

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

##### 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.

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. $$3 x+4 y=12$$

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