9.6 分流二次曲线

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  断离二次二次曲线

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  The general equation of a conic is $\begin{array}{r}A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0\end{array}$ .  This form is so general that it encompasses all regular lines, singular points and degenerate 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?
  ::二次曲线的一般方程式是Ax2+Bxy+Cy2+Dx+Ey+F=0。 这种形式非常笼统, 包括所有常规线条、 单点和像 X 一样的退化。 这是因为有几处特殊的例子说明平面如何交叉两个侧锥体。 这些退化形状是如何形成的 ?

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  Graphing Degenerate Conics
  ::绘制断层二次曲线图

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  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:
  ::退化的二次曲线是一种没有二次曲线通常特性的二次曲线。 退化的二次曲线方程式不能以图形形式写成。 退化的二次曲线有三种类型:

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  1. A singular point , which is of the form: $\begin{array}{r}\frac{(x-h{)}^2}{a}+\frac{(y-k{)}^2}{b}=0\end{array}$ . You can think of a singular point as a circle or an ellipse with an infinitely small radius. 
     ::单点, 即窗体的单点 x- h) 2a+(y- k) 2b=0。 您可以将单点视为圆或半径极小的椭圆或椭圆。
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  2. A line , which has coefficients  $\begin{array}{r}A=B=C=0\end{array}$ in the general equation of a conic.  The remaining portion of the equation is $\begin{array}{r}Dx+Ey+F=0\end{array}$ , which is a line.
     ::A 线,在二次曲线的一般方程式中含有系数A=B=C=0。方程式的剩余部分是Dx+Ey+F=0,这是一个直线。
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  3. A degenerate hyperbola , which is of the form:  $\begin{array}{r}\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=0\end{array}$ .  The result is two intersecting lines that make an “X” shape.  The slopes of the intersecting lines forming the X are $\begin{array}{r}\pm \frac{b}{a}\end{array}$ . This is because  $\begin{array}{r}b\end{array}$ goes with the  $\begin{array}{r}y\end{array}$ portion of the equation and is the rise, while  $\begin{array}{r}a\end{array}$ goes with the  $\begin{array}{r}x\end{array}$ portion of the equation and is the run.
     ::退化的双曲线, 其形状为 (x- h) 2a2- (y- k) 2b2=0。 结果是两条交叉线, 形成“ X” 形状。 构成 X 的交叉线的斜度为 ba。 这是因为 b 与方程的 y 部分相移, 是 上升, 而 a 则与 等式的 x 部分相移, 是运行 。

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  Examples
  ::实例

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  Example 1
  ::例1

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  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. 
  ::早些时候,有人问您如何形成腐蚀的二次曲线。当您用两侧锥体接触的双侧锥体交叉一架飞机时, 十字路口就是一个单一点。 当您用两侧锥体接触的双侧锥体交叉一架飞机, 以便让飞机接触一个锥体的边缘, 穿过中心点, 并继续接触另一个锥体的边缘时, 这会产生一条线。 当您用两侧锥体交叉一架飞机, 使飞机垂直通过两个锥体的中心点时, 它会产生一个退化的超大波。

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  Example 2
  ::例2

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  Transform the conic equation into standard form and sketch.
  ::将二次方程转换成标准形式和草图

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  $\begin{array}{r}0x^2+0xy+0y^2+2x+4y-6=0\end{array}$
  ::0x2+0xy+0y2+2x+4y-6=0

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  This is the line $\begin{array}{r}y=-\frac{1}{2}x+\frac{3}{2}\end{array}$ .
  ::这是y12x+32的线条。

  

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  Example 3
  ::例3

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  Transform the conic equation into standard form and sketch.
  ::将二次方程转换成标准形式和草图

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  $\begin{array}{r}3x^2-12x+4y^2-8y+16=0\end{array}$
  ::3x2 - 12x+4y2 - 8y+16=0

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  $\begin{array}{r}3x^2-12x+4y^2-8y+16=0\end{array}$
  ::3x2 - 12x+4y2 - 8y+16=0

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  $$\begin{array}{rl}3(x^2-4x)+4(y^2-2y)& =-16\\ 3(x^2-4x+4)+4(y^2-2y+1)& =-16+12+4\\ 3(x-2{)}^2+4(y-1{)}^2& =0\\ \frac{(x-2{)}^2}{4}+\frac{(y-1{)}^2}{3}& =0\end{array}$$

  
  ::3(x2-4x)+4(y2-2y)163(x2-4x+4)+4(y2-2y+1)+4(y2-2y+1) __16+12+43(x-2)2+4(y-1)2=0(x-2)24+(y-1)23=0

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  The point (2, 1) is the result of this degenerate conic.
  ::点(2,1)是这个堕落的二次曲线的结果。

  

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  Example 4
  ::例4

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  Transform the conic equation into standard form and sketch.
  ::将二次方程转换成标准形式和草图

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  $\begin{array}{r}16x^2-96x-9y^2+18y+135=0\end{array}$
  ::16x2-96x-9y2+18y+135=0

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  $\begin{array}{r}16x^2-96x-9y^2+18y+135=0\end{array}$
  ::16x2-96x-9y2+18y+135=0

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  $$\begin{array}{rl}16(x^2-6x)-9(y^2-2y)& =-135\\ 16(x^2-6x+9)-9(y^2-2y+1)& =-135+144-9\\ 16(x-3{)}^2-9(y-1{)}^2& =0\\ \frac{(x-3{)}^2}{9}-\frac{(y-1{)}^2}{16}& =0\end{array}$$

  
  ::16(x2-6x)-9(y2-2-2y)13516(x2-6x+9)-9(y2-2y+1)=135+144-916(x-33)2-9(y-1)2=0(x-3)29-(y-1)216=0

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  This is a degenerate hyperbola.
  ::这是一个堕落的双曲线。

  

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  Example 5
  ::例5

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  1. Create a conic that describes just the point (4, 7).
  ::1. 创建一个仅描述点的二次曲线(4,7)。

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  $\begin{array}{r}(x-4{)}^2+(y-7{)}^2=0\end{array}$
  :x-4)2+(y-7)2=0

  Summary

  播放段落 Degenerate conics are conics that do not have the usual properties and cannot be written in graphing form. There are three types: ::降解性二次曲线是没有通常特性的二次曲线,不能以图形形式写成。 播放段落 A singular point is a degenerate conic of the form $\begin{array}{r}\frac{(x-h{)}^2}{a}-\frac{(y-k{)}^2}{b}=0\end{array}$   ::单点是窗体( x-h) 2a-(y-k) 2b=0 的退化二次曲线 播放段落 A line when the coefficients $\begin{array}{r}A=B=C=0\end{array}$  in the general equation, resulting in the equation $\begin{array}{r}Dx+Ey+F=0.\end{array}$   ::当一般方程中的系数A=B=C=0导致公式Dx+Ey+F=0时的直线。 播放段落 A degenerate hyperbola is a degenerate conic of the form $\begin{array}{r}\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=0.\end{array}$  This looks like two lines in an “X”. ::退化的双曲线是形态(x-h)2a2-(y-k)2b2=0的退化二次曲线。 这看起来像“X”中的两行。

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  Review
  ::回顾

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  1. What are the three degenerate conics?
  ::1. 三种堕落的二次曲线是什么?

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  Change each equation into graphing form and state what type of conic or degenerate conic it is.
  ::将每个方程式改变为图形形式, 并显示它是哪种二次曲线或退化的二次曲线 。

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  2. $\begin{array}{r}{x}^2-6x-9y^2-54y-72=0\end{array}$
  ::2. x2-6x-9y2-54y-72=0

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  3. $\begin{array}{r}4x^2+16x-9y^2+18y-29=0\end{array}$
  ::3. 4x2+16x-9y2+18y-29=0

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  4. $\begin{array}{r}9x^2+36x+4y^2-24y+72=0\end{array}$
  ::4. 9x2+36x+4y2-24y+72=0

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  5. $\begin{array}{r}9x^2+36x+4y^2-24y+36=0\end{array}$
  ::5. 9x2+36x+4y2-24y+36=0

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  6. $\begin{array}{r}0x^2+5x+0y^2-2y+1=0\end{array}$
  ::6. 0x2+5x+0y2-2y+1=0

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  7. $\begin{array}{r}{x}^2+4x-y+8=0\end{array}$
  ::7. x2+4x-y+8=0

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  8. $\begin{array}{r}{x}^2-2x+y^2-6y+6=0\end{array}$
  ::8. x2-2x+y2-6y+6=0

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  9. $\begin{array}{r}{x}^2-2x-4y^2+24y-35=0\end{array}$
  ::9. x2-2x-4y2+24y-35=0

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  10. $\begin{array}{r}{x}^2-2x+4y^2-24y+33=0\end{array}$
  ::10. x2-2x+4y2-24y+33=0

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  Sketch each conic or degenerate conic.
  ::每一个曲线或腐烂的曲线

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  11. $\begin{array}{r}\frac{(x+2{)}^2}{4}+\frac{(y-3{)}^2}{9}=0\end{array}$
  ::11. (x+2)24+(y-3)29=0

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  12. $\begin{array}{r}\frac{(x-3{)}^2}{9}+\frac{(y+3{)}^2}{16}=1\end{array}$
  ::12. (x-3)29+(y+3)216=1

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  13. $\begin{array}{r}\frac{(x+2{)}^2}{9}-\frac{(y-1{)}^2}{4}=1\end{array}$
  ::13. (x+2)29-(y-1)24=1

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  14. $\begin{array}{r}\frac{(x-3{)}^2}{9}-\frac{(y+3{)}^2}{4}=0\end{array}$
  ::14. (x-3)29-(y+3)24=0

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  15. $\begin{array}{r}3x+4y=12\end{array}$
  ::15. 3x+4y=12

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  Review (Answers)
  ::回顾(答复)

  To see the answer key for this book, go to the and click on the Answer Key under the ' ' option.