节: 断离二次二次曲线 | 9.6 分流二次曲线

章节大纲

The general equation of a conic is $\begin{aligned} A x^{2} + B x y + C y^{2} + D x + E y + F = 0 \end{aligned}$ . 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 ?

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

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:

A singular point , which is of the form: $\begin{aligned} \frac{(x - h)^{2}}{a} + \frac{(y - k)^{2}}{b} = 0 \end{aligned}$ . 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。

A line , which has coefficients $\begin{aligned} A = B = C = 0 \end{aligned}$ in the general equation of a conic. The remaining portion of the equation is $\begin{aligned} D x + E y + F = 0 \end{aligned}$ , 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。

A degenerate hyperbola , which is of the form: $\begin{aligned} \frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 0 \end{aligned}$ . The result is two intersecting lines that make an “X” shape. The slopes of the intersecting lines forming the X are $\begin{aligned} \pm \frac{b}{a} \end{aligned}$ . This is because $\begin{aligned} b \end{aligned}$ goes with the $\begin{aligned} y \end{aligned}$ portion of the equation and is the rise, while $\begin{aligned} a \end{aligned}$ goes with the $\begin{aligned} x \end{aligned}$ 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。

Examples
::实例

Example 1
::例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.
::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。

Example 2
::例2

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

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

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

Example 3
::例3

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

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

\begin{aligned} 3 (x^{2} - 4 x) + 4 (y^{2} - 2 y) & = - 16 \\ 3 (x^{2} - 4 x + 4) + 4 (y^{2} - 2 y + 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{aligned}

::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

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。

Example 4
::例4

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

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

\begin{aligned} 16 (x^{2} - 6 x) - 9 (y^{2} - 2 y) & = - 135 \\ 16 (x^{2} - 6 x + 9) - 9 (y^{2} - 2 y + 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{aligned}

::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

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

Example 5
::例5

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)。

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

Summary

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。

A singular point is a degenerate conic of the form $\begin{aligned} \frac{(x - h)^{2}}{a} - \frac{(y - k)^{2}}{b} = 0 \end{aligned}$
::单点是窗体( x-h) 2a-(y-k) 2b=0 的退化二次曲线

A line when the coefficients $\begin{aligned} A = B = C = 0 \end{aligned}$ in the general equation, resulting in the equation $\begin{aligned} D x + E y + F = 0. \end{aligned}$
::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。

A degenerate hyperbola is a degenerate conic of the form $\begin{aligned} \frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 0. \end{aligned}$ 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。

Review
::回顾

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

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。

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Review (Answers)
::回顾(答复)

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章节大纲

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

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

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

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)