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# Degenerate Conics

## Point, line, or pair of lines formed when some coefficients of a conic equal zero.

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Degenerate Conics

The general equation of a conic is Ax2+Bxy+Cy2+Dx+Ey+F=0\begin{align*}Ax^2+Bxy+Cy^2+Dx+Ey+F=0\end{align*}.  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

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: (xh)2a+(yk)2b=0\begin{align*}\frac{(x-h)^2}{a}+\frac{(y-k)^2}{b}=0\end{align*}. 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\begin{align*}A=B=C=0\end{align*} in the general equation of a conic.  The remaining portion of the equation is Dx+Ey+F=0\begin{align*}Dx+Ey+F=0\end{align*}, which is a line.
3. A degenerate hyperbola, which is of the form:  (xh)2a(yk)2b=0\begin{align*}\frac{(x-h)^2}{a}-\frac{(y-k)^2}{b}=0\end{align*}.  The result is two intersecting lines that make an “X” shape.  The slopes of the intersecting lines forming the X are ±ba\begin{align*}\pm \frac{b}{a}\end{align*}. This is because b\begin{align*}b\end{align*} goes with the y\begin{align*}y\end{align*} portion of the equation and is the rise, while a\begin{align*}a\end{align*} goes with the x\begin{align*}x\end{align*} 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.

0x2+0xy+0y2+2x+4y6=0\begin{align*}0x^2+0xy+0y^2+2x+4y-6=0\end{align*}

This is the line y=12x+32\begin{align*}y=-\frac{1}{2} x+\frac{3}{2}\end{align*}.

#### Example 3

Transform the conic equation into standard form and sketch.

3x212x+4y28y+16=0\begin{align*}3x^2-12x+4y^2-8y+16=0\end{align*}

3x212x+4y28y+16=0\begin{align*}3x^2-12x+4y^2-8y+16=0 \end{align*}

3(x24x)+4(y22y)3(x24x+4)+4(y22y+1)3(x2)2+4(y1)2(x2)24+(y1)23=16=16+12+4=0=0\begin{align*}3(x^2-4x)+4(y^2-2y)&=-16\\ 3(x^2-4x+4)+4(y^2-2y+1)&=-16+12+4\\ 3(x-2)^2+4(y-1)^2&=0\\ \frac{(x-2)^2}{4}+\frac{(y-1)^2}{3}&=0 \end{align*}

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

#### Example 4

Transform the conic equation into standard form and sketch.

16x296x9y2+18y+135=0\begin{align*}16x^2-96x-9y^2+18y+135=0\end{align*}

16x296x9y2+18y+135=0\begin{align*}16x^2-96x-9y^2+18y+135=0\end{align*}

16(x26x)9(y22y)16(x26x+9)9(y22y+1)16(x3)29(y1)2(x3)29(y1)216=135=135+1449=0=0\begin{align*}16(x^2-6x)-9(y^2-2y)&=-135\\ 16(x^2-6x+9)-9(y^2-2y+1)&=-135+144-9\\ 16(x-3)^2-9(y-1)^2&=0\\ \frac{(x-3)^2}{9}-\frac{(y-1)^2}{16}&=0\end{align*}

This is a degenerate hyperbola.

#### Example 5

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

(x4)2+(y7)2=0\begin{align*}(x-4)^2+(y-7)^2=0\end{align*}

### 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. x26x9y254y72=0\begin{align*}x^2-6x-9y^2-54y-72=0\end{align*}

3. 4x2+16x9y2+18y29=0\begin{align*}4x^2+16x-9y^2+18y-29=0\end{align*}

4. 9x2+36x+4y224y+72=0\begin{align*}9x^2+36x+4y^2-24y+72=0\end{align*}

5. 9x2+36x+4y224y+36=0\begin{align*}9x^2+36x+4y^2-24y+36=0\end{align*}

6. 0x2+5x+0y22y+1=0\begin{align*}0x^2+5x+0y^2-2y+1=0\end{align*}

7. x2+4xy+8=0\begin{align*}x^2+4x-y+8=0\end{align*}

8. x22x+y26y+6=0\begin{align*}x^2-2x+y^2-6y+6=0\end{align*}

9. x22x4y2+24y35=0\begin{align*}x^2-2x-4y^2+24y-35=0\end{align*}

10. x22x+4y224y+33=0\begin{align*}x^2-2x+4y^2-24y+33=0\end{align*}

Sketch each conic or degenerate conic.

11. (x+2)24+(y3)29=0\begin{align*}\frac{(x+2)^2}{4}+\frac{(y-3)^2}{9}=0 \end{align*}

12. (x3)29+(y+3)216=1\begin{align*}\frac{(x-3)^2}{9}+\frac{(y+3)^2}{16}=1 \end{align*}

13. (x+2)29(y1)24=1\begin{align*}\frac{(x+2)^2}{9}-\frac{(y-1)^2}{4}=1 \end{align*}

14. (x3)29(y+3)24=0\begin{align*}\frac{(x-3)^2}{9}-\frac{(y+3)^2}{4}=0 \end{align*}

15. 3x+4y=12\begin{align*}3x+4y=12\end{align*}

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

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### Vocabulary Language: English

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.