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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 \begin{align*}Ax^2+Bxy+Cy^2+Dx+Ey+F=0\end{align*}Ax2+Bxy+Cy2+Dx+Ey+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

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: \begin{align*}\frac{(x-h)^2}{a}+\frac{(y-k)^2}{b}=0\end{align*}(xh)2a+(yk)2b=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 \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.
  3. A degenerate hyperbola, which is of the form:  \begin{align*}\frac{(x-h)^2}{a}-\frac{(y-k)^2}{b}=0\end{align*}(xh)2a(yk)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.

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.

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

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

Example 3

Transform the conic equation into standard form and sketch.

\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*}3x212x+4y28y+16=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*}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

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

Example 4

Transform the conic equation into standard form and sketch.

\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*}16x296x9y2+18y+135=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*}16(x26x)9(y22y)16(x26x+9)9(y22y+1)16(x3)29(y1)2(x3)29(y1)216=135=135+1449=0=0

This is a degenerate hyperbola.

Example 5

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

\begin{align*}(x-4)^2+(y-7)^2=0\end{align*}(x4)2+(y7)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. \begin{align*}x^2-6x-9y^2-54y-72=0\end{align*}x26x9y254y72=0

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

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

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

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

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

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

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

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

Sketch each conic or degenerate conic.

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

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

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

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

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

Review (Answers)

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

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Vocabulary

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.

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