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Degenerate Conics

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Degenerate Conics
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The general equation of a conic is Ax^2+Bxy+Cy^2+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?

Guidance

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 Dx+Ey+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.

Example A

Transform the conic equation into standard form and sketch.

0x^2+0xy+0y^2+2x+4y-6=0

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

Example B

Transform the conic equation into standard form and sketch.

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

Solution: 3x^2-12x+4y^2-8y+16=0

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

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

Example C

Transform the conic equation into standard form and sketch.

16x^2-96x-9y^2+18y+135=0

Solution:   16x^2-96x-9y^2+18y+135=0

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

This is a degenerate hyperbola.

Concept Problem Revisited

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

Vocabulary

A degenerate conic is a conic that does not have the usual properties of a conic.  Since some of the coefficients of the general equation are zero, the basic shape of the conic is merely a point, a line or a pair of lines.  The connotation of the word degenerate means that the new graph is less complex than the rest of conics.

Guided Practice

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

2. Transform the conic equation into standard form and sketch.

-4x^2+8x+y^2+4y=0

3. Can you tell just by looking at a conic in general form if it is a degenerate conic?

Answers:

1. (x-4)^2+(y-7)^2=0

2.

-4x^2+8x+y^2+4y&=0\\-4(x^2-2x)+(y^2+4y)&=0\\-4(x^2-2x+1)+(y^2+4y+4)&=-4+4\\-4(x-1)^2+(y+2)^2&=0\\\frac{(x-1)^2}{1}-\frac{(y+2)^2}{4}&=0

3. In general you cannot tell if a conic is degenerate from the general form of the equation.  You can tell that the degenerate conic is a line if there are no  x^2 or y^2  terms, but other than that you must always try to put the conic equation into graphing form and see whether it equals zero because that is the best way to identify degenerate conics.

Practice

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^2-6x-9y^2-54y-72=0

3. 4x^2+16x-9y^2+18y-29=0

4. 9x^2+36x+4y^2-24y+72=0

5. 9x^2+36x+4y^2-24y+36=0

6. 0x^2+5x+0y^2-2y+1=0

7. x^2+4x-y+8=0

8. x^2-2x+y^2-6y+6=0

9. x^2-2x-4y^2+24y-35=0

10. x^2-2x+4y^2-24y+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. 3x+4y=12

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