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6.5.3: Degenerate Conics

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    14770
  • 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 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

    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 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.

    Solution

    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.

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

    Solution

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

    f-d_7e8667cd80d543ecc94a05f3c351d77aa5d3cf704a791fe6adf2fd99+IMAGE_THUMB_POSTCARD_TINY+IMAGE_THUMB_POSTCARD_TINY.png

    Example 3

    Transform the conic equation into standard form and sketch.

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

    Solution

    \(\ \begin{aligned}
    3 x^{2}-12 x+4 y^{2}-8 y+16&=0  \\
    3\left(x^{2}-4 x\right)+4\left(y^{2}-2 y\right) &=-16 \\
    3\left(x^{2}-4 x+4\right)+4\left(y^{2}-2 y+1\right) &=-16+12+4 \\
    3(x-2)^{2}+4(y-1)^{2} &=0 \\
    \frac{(x-2)^{2}}{4}+\frac{(y-1)^{2}}{3} &=0
    \end{aligned}\)

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

    f-d_3000d08d9afd24108b4f58f18d3a2e0b38d44a71dbb721d5eae9e094+IMAGE_THUMB_POSTCARD_TINY+IMAGE_THUMB_POSTCARD_TINY.png

    Example 4

    Transform the conic equation into standard form and sketch.

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

    Solution

    \(\ \begin{aligned}
    16 x^{2}-96 x-9 y^{2}+18 y+135=0 & \\
    16\left(x^{2}-6 x\right)-9\left(y^{2}-2 y\right) &=-135 \\
    16\left(x^{2}-6 x+9\right)-9\left(y^{2}-2 y+1\right) &=-135+144-9 \\
    16(x-3)^{2}-9(y-1)^{2} &=0 \\
    \frac{(x-3)^{2}}{9}-\frac{(y-1)^{2}}{16} &=0
    \end{aligned}\)

    This is a degenerate hyperbola.

    f-d_1020c4e8aa8f5fece6ca7560cb3c98fd769455e9431a8fc5444f2363+IMAGE_THUMB_POSTCARD_TINY+IMAGE_THUMB_POSTCARD_TINY.png

    Example 5

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

    Solution

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


    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.

    1. \(\ x^{2}-6 x-9 y^{2}-54 y-72=0\)
    2. \(\ 4 x^{2}+16 x-9 y^{2}+18 y-29=0\)
    3. \(\ 9 x^{2}+36 x+4 y^{2}-24 y+72=0\)
    4. \(\ 9 x^{2}+36 x+4 y^{2}-24 y+36=0\)
    5. \(\ 0 x^{2}+5 x+0 y^{2}-2 y+1=0\)
    6. \(\ x^{2}+4 x-y+8=0\)
    7. \(\ x^{2}-2 x+y^{2}-6 y+6=0\)
    8. \(\ x^{2}-2 x-4 y^{2}+24 y-35=0\)
    9. \(\ x^{2}-2 x+4 y^{2}-24 y+33=0\)

    Sketch each conic or degenerate conic.

    1. \(\ \frac{(x+2)^{2}}{4}+\frac{(y-3)^{2}}{9}=0\)
    2. \(\ \frac{(x-3)^{2}}{9}+\frac{(y+3)^{2}}{16}=1\)
    3. \(\ \frac{(x+2)^{2}}{9}-\frac{(y-1)^{2}}{4}=1\)
    4. \(\ \frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{4}=0\)
    5. \(\ 3 x+4 y=12\)

    Review (Answers)

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


    Vocabulary

    Term Definition
    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.

    Image Attributions

    1. [Figure 1]
      Credit: CK-12 Foundation
      License: CC BY-SA
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