Section: 断离二次二次曲线 | 11.7 离子二次曲线

Section outline

Introduction
::导言

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 and curves, singular points, and degenerate that look like an X. T here are a few special cases of how a plane can intersect a 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 。这种形式过于笼统, 包括了所有常规线条和曲线、单点, 以及像 X 一样的堕落。在平面如何交叉锥形方面, 有一些特殊的例子。这些退化形状是如何形成的 ?

Degenerate Conics
::断离二次二次曲线

A degenerate conic is generated when a plane intersects the vertex of the cone . There are three types of degenerate conics:
::当平面交叉锥体的顶部时,就会产生退化的二次曲线。有三种类型的退化的二次曲线:

The degenerate form of a circle or an ellipse is a singular point . At the vertex of the cone, the radius is 0, $\begin{aligned} r = 0 \end{aligned}$ . Thus, t he standard equation is $\begin{aligned} \frac{(x - h)^{2}}{a} + \frac{(y - k)^{2}}{b} = 0 \end{aligned}$ .
::圆或椭圆的退化形式是一个单点。在锥体的顶端,半径为 0, r=0。因此, 标准方程式是 (x-h) 2a+(y-k) 2b=0 。

The degenerate form of a parabola is a line or two parallel lines . For this conic section, the coefficients $\begin{aligned} A = B = C = 0 \end{aligned}$ in the general equation. Thus, the r esulting general equation is $\begin{aligned} D x + E y + F = 0 \end{aligned}$ .
::抛物线的退化形式是一条线或两条平行线。对于此二次曲线部分,一般方程中的系数A=B=C=0。因此,由此产生的一般方程式是Dx+Ey+F=0。

The degenerate form of a hyperbola is two intersecting lines . A t the vertex of the cone, $\begin{aligned} r = 0 \end{aligned}$ , so the standard equation is $\begin{aligned} \frac{(x - h)^{2}}{a} - \frac{(y - k)^{2}}{b} = 0 \end{aligned}$ . Thus, the slopes of the intersecting lines are $\begin{aligned} m = \pm \frac{b}{a}, \end{aligned}$ because $\begin{aligned} b \end{aligned}$ corresponds to the $\begin{aligned} y \end{aligned}$ portion of the equation, and $\begin{aligned} a \end{aligned}$ corresponds to the $\begin{aligned} x \end{aligned}$ portion of the equation.
::双倍波拉的退化形式是两个交叉线。在锥体的顶端, r=0, 所以标准方程是 (x-h) 2a- (y-k) 2b=0。因此, 交叉线的斜度是 mba, 因为 b 对应于方程的 y 部分, 和 a 对应方程的 x 部分。

The following video reviews the four : circle, ellipse, parabola, and hyperbola:
::以下视频回顾这四个方面:圆圈、椭圆、抛物线和双曲线:

Examples
::实例

Example 1
::例1

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

Solution:
::解决方案 :

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

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

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

::3x2 - 12x+4y2 - 8y+16=0

Solution:
::解决方案 :

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

::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 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

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

::16x2-96x-9y2+18y+135=0

Solution:
::解决方案 :

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

::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 4
::例4

Recall the question from the Introduction: H ow are these degenerate shapes formed?
::回顾导言中的问题:这些退化的形状是如何形成的?

Solution:
::解决方案 :

When you intersect a plane with a cone at the vertex where the two cones touch, the intersection is a single point. When you intersect a plane with the edge of one cone, passing through the vertex point, and continuing to touch the edge of the other conic, this produces a line or two parallel lines. When you intersect a plane with a cone so that the plane passes vertically through the vertex , it produces two intersecting lines.
::当您在两个锥体接触的顶端交叉一个平面和锥体的锥形交叉时,该十字路口是一个点。当您在一个锥体边缘交叉一个平面,穿过顶端点,继续接触另一锥体的边缘时,它会产生一条线或两条平行线。当您用锥体交叉一个平面,使平面垂直通过顶端时,它产生两条交叉线。

Example 5
::例5

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

Solution:
::解决方案 :

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

:x-4)2+(y-7)2=0

Example 6
::例6

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

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

::− 4x2+8x+y2+4y=0

Solution:
$\begin{aligned} - 4 x^{2} + 8 x + y^{2} + 4 y & = 0 \\ - 4 (x^{2} - 2 x) + (y^{2} + 4 y) & = 0 \\ - 4 (x^{2} - 2 x + 1) + (y^{2} + 4 y + 4) & = - 4 + 4 \\ - 4 (x - 1)^{2} + (y + 2)^{2} & = 0 \\ \frac{(x - 1)^{2}}{1} - \frac{(y + 2)^{2}}{4} & = 0 \end{aligned}$

::溶解度:-4x2+8x+8x+y2+4y=0-4(x2-2x)+(y2+4y)=0-4(x2-2x+1)+(y2+4+4)+4+4-4(y2+4)+4-4(x-1)2+(y+2)2=0(x-1)21-21-(y+2)24=0

Example 7
::例7

How can you tell just by looking at a conic in general form if it is a degenerate conic?
::如果它是堕落的二次曲线,你怎么能仅仅看一般的二次曲线来判断呢?

Solution:
::解决方案 :

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 $\begin{aligned} x^{2} \end{aligned}$ or $\begin{aligned} y^{2} \end{aligned}$ terms. However, you should always try to put the conic equation into graphing form to see whether it equals zero, because that is the best way to identify degenerate conics.
::一般来说, 您无法辨别二次曲线是否从方程式的一般形式中退化。您可以辨别, 如果没有 x2 或 y2 条件, 则二次曲线是一条线。但是, 您应该总是尝试将二次曲线的方程式放入图形形式, 以确定它是否等于零, 因为这是识别二次曲线的最好方法。

Summary
::摘要

A degenerate conic is generated when a plane intersects the vertex of the cone .
::当平面交叉锥体的顶部时,就会产生退化的二次曲线。

There are three types of degenerate conics: a single point, a line or two parallel lines, or two intersecting lines.
::有三种退化的二次曲线:一点、一线或两条平行线,或两条交叉线。

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:
::将每个方程式修改为图形化形式, 并声明它是哪种二次曲线或退化二次曲线 :

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:
::每个二次曲线或退化的二次曲线:

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)

Please see the Appendix.
::请参看附录。

Section outline

Introduction ::导言

Degenerate Conics ::断离二次二次曲线

Examples ::实例

Summary ::摘要

Review ::回顾

Review (Answers ) ::回顾(答复)

Introduction
::导言

Degenerate Conics
::断离二次二次曲线

Examples
::实例

Summary
::摘要

Review
::回顾

Review (Answers )
::回顾(答复)