Section: 二次曲线的一般形态 | 11.2 二次曲线的一般形式

Section outline

Introduction
::导言

Conics are a family of graphs that include , circles, , and . All of these graphs are derived from the same general equation. By manipulating this specific equation, you can determine the type of conic and graph it, using key information.
::二次曲线是包含、圆圈、和的图形组合。所有这些图形都来自相同的普通方程式。通过操纵这个特定的方程式, 您可以使用关键信息来决定二次曲线的类型和图形。

Conics
::二次二次曲线

The word "conic" comes from the word "cone," which is where the shapes of parabolas, circles, ellipses, and hyperbolas originate. Consider a double cone, defined as the form generated when one of two intersecting lines is rotated about the other. The lines that pass through the vertex to form this cone are called the generators . The fixed line corresponding to the height of the cone is called the axis of the cone. are the nondegenerate curves generated by intersecting one or both pieces of the cone with a plane. One piece of a double cone is called a nappe.
::“ 二次曲线” 一词来自“ 锥形” 一词, 锥形、圆形、椭圆形和双螺旋体的形状来源于此。考虑一个双锥体, 定义为两个交叉线之一在另一端旋转时生成的形态。穿过顶端形成此锥体的线被称为“ 生成器 ” 。与锥形高度相对应的固定线被称为锥形的轴。是将锥形的一个或两个部分与平面交叉产生的非降解曲线。双锥体的一块称为环形。

When a plane is perpendicular to the axis of the cone, a circle is generated.
::当一平面与锥体轴垂直时,即产生圆圈。

When a plane is not perpendicular to the axis of the cone but does intersect one nappe of the cone, either an ellipse or a parabola is generated. An ellipse is generated when the plane is tilted so that it intersects both generator lines.
::当一架飞机与锥体轴不垂直,但与锥体角的一个环形交叉时,即产生椭圆或抛物线。当平面倾斜时,即产生椭圆,使两条发电机线交叉。

A parabola is generated when the plane is tilted so that it is parallel to one generator and intersects the other generator.
::当飞机倾斜时产生抛物线,使其与一个发电机平行,并交叉另一个发电机。

When a plane intersects both nappes of the cone, a hyperbola is generated.
::当一平面交叉两个锥形腺时,会产生双倍波拉。

In the coordinate plane, each of the above conics can be graphed using the same general formula and by applying important information about the size and place of the points.
::在坐标平面上,上述每一个二次曲线都可以用相同的通用公式和通过应用关于点大小和位置的重要信息来绘制图表。

Standard F orm of a Conic
::二次曲线的标准窗体

$\begin{align*}Ax^2+Bxy+Cy^2+Dx+Ey+F=0\end{align*}$

In this form, examining the two coefficients $\begin{align*}A\end{align*}$ and $\begin{align*}C\end{align*}$ will expose the type of conic.
::以这种形式审查两个系数A和系数C将暴露这种二次曲线的类型。

For circles, the coefficients of $\begin{align*}x^2\end{align*}$ and $\begin{align*}y^2\end{align*}$ are the same sign and the same value: $\begin{align*}A=C.\end{align*}$
::对于圆圈,x2和y2的系数是相同的符号和相同值:A=C。

For ellipses, the coefficients of $\begin{align*}x^2\end{align*}$ and $\begin{align*}y^2\end{align*}$ are the same sign and different values: $\begin{align*}A, C > 0, \ A \neq C.\end{align*}$
::对于省略号,x2和y2的系数是相同的符号和不同的值:A,C>0,AC。

For hyperbolas, the coefficients of $\begin{align*}x^2\end{align*}$ and $\begin{align*}y^2\end{align*}$ are opposite signs: $\begin{align*}C < 0 < A\end{align*}$ or $\begin{align*}A < 0 < C.\end{align*}$
::对于双光蜡,x2和y2的系数是相反的符号:C<0<A或A<0<C。

For parabolas, either the coefficient of $\begin{align*}x^2\end{align*}$ or $\begin{align*}y^2\end{align*}$ must be zero: $\begin{align*}A=0\end{align*}$ or $\begin{align*}C=0.\end{align*}$
::对于parabolas, x2 或 y2 的系数必须为零: A=0 或 C=0。

Each specific type of conic has its own graphing form, but in all cases the technique of completing the square is essential. The examples below review completing the square and recognizing conics.
::每种具体类型的二次曲线都有其自己的图形形式,但在所有情况下,完成方形的技术都是必不可少的,下文举例说明了完成方形和确认二次曲线的情况。

The following video demonstrates how you can generate a circle, ellipse, parabola, and hyperbola by intersecting a cone with a plane:
::以下影片展示了如何通过将锥形与平面交叉而产生圆圈、椭圆、抛物线和双波形:

Examples
::实例

Example 1
::例1

Complete the square in the expression $\begin{align*}x^2+6x\end{align*}$ . Demonstrate graphically what completing the square represents.
::在表达式 x2+6x 中填全正方形。以图形方式显示正方形代表的完整内容。

Solution:
::解决方案 :

Algebraically, completing the square just requires you to divide the coefficient of $\begin{align*}x\end{align*}$ by 2 and square the result. In this case, $\begin{align*}\left(\frac{6}{2}\right)^2=3^2=9\end{align*}$ . Since you cannot add 9 to an expression without changing its value, you must simultaneously add 9 and subtract 9 so the net change will be zero.
::代数, 完成正方块只需要您将x系数除以 2 和结果平方。在这种情况下, (622=32=9) 。由于您不能在不改变表达式值的情况下添加 9, 因此您必须同时添加 9 和减去 9, 这样净变化将是零。

$\begin{aligned} x^{2} + 6 x + 9 - 9 \end{aligned}$
$\begin{align*}x^2+6x+9-9\end{align*}$
::x2+6x+9-9

Now you can factor by recognizing a perfect square.
::现在您可以通过识别一个完美的正方块来考虑因素。

$\begin{aligned} (x + 3)^{2} - 9 \end{aligned}$
$\begin{align*}(x+3)^2-9\end{align*}$
:x+3)2-9

Graphically, the original expression $\begin{align*}x^2+6x\end{align*}$ can be represented by the area of a rectangle with sides $\begin{align*}x\end{align*}$ and $\begin{align*}(x+6)\end{align*}$ .
::图形化时,原表达式 x2+6x 可以用侧形 x 和 (x+6) 的矩形区域表示。

The term "complete the square" has visual meaning as well algebraic meaning. The rectangle can be rearranged to be more square-like, so that instead of small rectangle of area $\begin{align*}6x\end{align*}$ at the bottom, there is a rectangle of area $\begin{align*}3x\end{align*}$ on two sides of the $\begin{align*}x^2\end{align*}$ square.
::“ 完成正方” 一词具有视觉意义和代数含义。矩形可以重新排列为更像正方形, 这样, x2 方的两侧没有在下方区域 6x 上的小矩形, 反而有一个区域 3x 的矩形。

W hat is missing to complete this shape as a perfect square? A little corner square of 9 is missing, which is why the 9 should be added to make the perfect square of $\begin{align*}(x+3)(x+3)\end{align*}$ .
::完成这个完美的正方形缺少什么? 缺少一个小角方块9的正方块, 正因为如此,才应该增加9个方块,使(x+3)(x+3)+3的正方块成为完美的正方块。

Example 2
::例2

What type of conic is each of the following relations?
::下列关系中,每种是哪种二次曲线?

$\begin{align*}5y^2-2x^2=-25\end{align*}$
::5y2-2x225

$\begin{align*}x=-\frac{1}{2}y^2-3\end{align*}$
::x12y2 - 3

$\begin{align*}4x^2+6y^2=36\end{align*}$
::4x2+6y2=36

$\begin{align*}x^2-\frac{1}{4}y=1\end{align*}$
::x2 - 14y=1

$\begin{align*}-\frac{x^2}{8}+\frac{y^2}{4}=1\end{align*}$
::-x28+y24=1

$\begin{align*}-x^2+99y^2=12\end{align*}$
::-x2+99y2=12

Solutions:
::解决办法:

Hyperbola because the $\begin{align*}x^2\end{align*}$ and $\begin{align*}y^2\end{align*}$ coefficients are different signs.
::Hyperbola 因为 x2 和 y2 系数是不同的符号。

Parabola (sideways) because the $\begin{align*}x^2\end{align*}$ term is missing.
::parabola( 边缘) , 因为 X2 术语缺失。

Ellipse because the $\begin{align*}x^2\end{align*}$ and $\begin{align*}y^2\end{align*}$ coefficients are different values but the same sign.
::椭圆, 因为 x2 和 y2 系数是不同的值, 但符号相同。

Parabola (upright) because the $\begin{align*}y^2\end{align*}$ term is missing.
::parabola (直截了当) , 因为 Y2 术语缺失。

Hyperbola because the $\begin{align*}x^2\end{align*}$ and $\begin{align*}y^2\end{align*}$ coefficients are different signs.
::Hyperbola 因为 x2 和 y2 系数是不同的符号。

Hyperbola because the $\begin{align*}x^2\end{align*}$ and $\begin{align*}y^2\end{align*}$ coefficients are different signs.
::Hyperbola 因为 x2 和 y2 系数是不同的符号。

Example 3
::例3

Complete the square for both the $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ terms in the equation below.
::以下方程式中的 x 和 y 条件填充方形。

$\begin{aligned} x^{2} + 6 x + 2 y^{2} + 16 y = 0 \end{aligned}$
$\begin{align*}x^2+6x+2y^2+16y=0\end{align*}$
::x2+6x+2y2+16y=0

Solution:
::解决方案 :

First, write out the equation with space, so there is room for the terms to be added to both sides. Since this is an equation, it is appropriate to add the values to both sides instead of adding and subtracting the same value simultaneously. As you rewrite with spaces, factor out any coefficient of the $\begin{align*}x^2\end{align*}$ or $\begin{align*}y^2\end{align*}$ terms, since your algorithm for completing the square works only when this coefficient is 1.
::首先,用空格写出方程式, 这样可以将条件添加到两边。由于这是一个方程式, 将数值添加到两边是合适的, 而不是同时增减相同值。当你用空格重写时, 请将 X2 或 y2 条件的任何系数除掉, 因为只有在这个系数为 1 时, 您完成方块的算法才会起作用。

$\begin{aligned} x^{2} + 6 x + \underline{} + 2 (y^{2} + 8 y + \underline{}) = 0 \end{aligned}$
$\begin{align*}x^2+6x+ \underline{\;\;\;\;\;}+2(y^2+8y+\underline{\;\;\;\;\;})=0\end{align*}$
::x2+6x 2(y2+8y)=0

Next, complete the square by adding a 9 and what looks like a 16 on the left. (It is actually a 32, since it is inside the parentheses.)
::其次,通过在左侧添加一个 9 和看起来像一个 16 来完成方形。 (实际上是32 , 因为它在括号内。 )

$\begin{aligned} x^{2} + 6 x + 9 + 2 (y^{2} + 8 y + 16) = 9 + 32 \end{aligned}$
$\begin{align*}x^2+6x+9+2(y^2+8y+16)=9+32\end{align*}$
::x2+6x+9+9+2(y2+8y+16)=9+32

Factor.
::因素。

$\begin{aligned} (x + 3)^{2} + 2 (y + 4)^{2} = 41 \end{aligned}$
$\begin{align*}(x+3)^2+2(y+4)^2=41\end{align*}$
:x+3)2+2(y+4)2=41

Example 4
::例4

Identify the type of conic in each of the following relations:
::确定下列关系中每种关系中的二次曲线类型:

$\begin{align*}3x^2=3y^2+18\end{align*}$
::3x2=3y2+18 3x2=3y2+18

$\begin{align*}y=4(x-3)^2+2\end{align*}$
::y=4(x-3)2+2

$\begin{align*}x^2+y^2=4\end{align*}$
::x2+y2=4 x2+y2=4

$\begin{align*}y^2+2y+x^2-6x=12\end{align*}$
::y2y+x2 - 6x=12

$\begin{align*}\frac{x^2}{6}+\frac{y^2}{12}=1\end{align*}$
::x26+y212=1

$\begin{align*}x^2-y^2+4=0\end{align*}$
::x2-y2+4=0

Solutions:
::解决办法:

1. The relation is a hyperbola because when you move the $\begin{align*}3y^2\end{align*}$ to the lefthand side of the equation, it becomes negative, and then the coefficients of $\begin{align*}x^2\end{align*}$ and $\begin{align*}y^2\end{align*}$ have opposite signs.
::1. 关系是双曲线,因为当将3y2移到方程左侧时,它变成负值,然后x2和y2的系数有相反的符号。

2. Parabola
::2. 帕拉波拉

3. Circle
::3. 圆环

4. Circle
::4. 圆环

5. Ellipse
::5. 椭圆

6. Hyperbola
::6. 超重波

Example 5
::例5

Complete the square in the following expression:
$\begin{aligned} 6 y^{2} - 36 y + 4. \end{aligned}$
$\begin{align*}6y^2-36y+4.\end{align*}$
::以下列表达式: 6y2- 36y+4 填写正方形。

Solution:
::解决方案 :

$\begin{aligned} 6 y^{2} - 36 y + 4 \\ 6 (y^{2} - 6 y + \underline{}) + 4 \\ 6 (y^{2} - 6 y + 9) + 4 - 54 \\ 6 (y - 3)^{2} - 50 \end{aligned}$
$\begin{align*}& 6y^2-36y+4\\ & 6 (y^2-6y+ \underline{\;\;\;\;})+4\\ & 6(y^2-6y+9)+4-54\\ & 6(y-3)^2-50\end{align*}$
::6y2-36y+46(y2-6y)+46(y2-6y+9)+4-546(y-3)2-50

Example 6
::例6

Complete the square for both $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ in the equation below.
$\begin{aligned} - 3 x^{2} - 24 x + 4 y^{2} - 32 y = 8 \end{aligned}$
$\begin{align*}-3x^2-24x+4y^2-32y=8\end{align*}$
::在以下方程中, x 和 y 的 x 和 y 填全方形。- 3x2 - 24x+4y2 - 32y2 - 32y=8

Solution:
::解决方案 :

$\begin{aligned} - 3 x^{2} - 24 x + 4 y^{2} - 32 y & = 8 \\ - 3 (x^{2} + 8 x + \underline{}) + 4 (y^{2} - 8 y + \underline{}) & = 8 \\ - 3 (x^{2} + 8 x + 16) + 4 (y^{2} - 8 y + 16) & = 8 - 48 + 64 \\ - 3 (x + 4)^{2} + 4 (y - 4)^{2} & = 24 \end{aligned}$
$\begin{align*}-3x^2-24x+4y^2-32y &= 8\\ -3(x^2+8x+ \underline{\;\;\;\;})+4(y^2-8y+\underline{\;\;\;\;}) &= 8\\ -3 (x^2+8x+16)+4(y^2-8y+16) &= 8-48+64\\ -3 (x+4)^2+4 (y-4)^2 &= 24\end{align*}$
::3x2-24x+4y2-32y=8-3(x2+8x)+4(y2-8y)=8-3(x2+8x+16)+4(y2-8y+16)+4(y2-8y+16)=8-48-64-3(x+4)2+4(y-4)2=24

Summary
::摘要

Conic sections are the nondegenerate curves generated by intersecting one or both pieces of the cone with a plane.
::二次曲线系通过将锥体的一个或两个部分与平面交叉而生成的非变性曲线。

There are four conic sections: circles, ellipses, parabolas, and hyperbolas.
::共有四个二次曲线:圆圈、椭圆、parapolas和超二次曲线。

Standard form of a Conic : $\begin{align*}Ax^2+Bxy+Cy^2+Dx+Ey+F=0\end{align*}$
::二次曲线的标准窗体:Ax2+Bxy+Cy2+Dx+Ey+F=0

Review
::回顾

Identify the type of conic in each of the following relations:
::确定下列关系中每种关系中的二次曲线类型:

1. $\begin{align*}3x^2+4y^2=12\end{align*}$
::1. 3x2+4y2=12

2. $\begin{align*}x^2+y^2=9\end{align*}$
::2. x2+y2=9

3. $\begin{align*}\frac{x^2}{4}+\frac{y^2}{9}=1\end{align*}$
::3. x24+y29=1

4. $\begin{align*}y^2+x=11\end{align*}$
::4. y2+x=11

5. $\begin{align*}x^2+2x-y^2+6y=15\end{align*}$
::5. x2+2x-y2+6y=15

6. $\begin{align*}x^2=y-1\end{align*}$
::6. x2=y- 1

Complete the square for $\begin{align*}x\end{align*}$ or $\begin{align*}y\end{align*}$ in each of the following expressions:
::下列表达式中的 x 或 y , 填全正方形 :

7. $\begin{align*}x^2+4x\end{align*}$
::7. x2+4x

8. $\begin{align*}y^2-8y\end{align*}$
::8.y2-8 y2-8y

9. $\begin{align*}3x^2+6x+4\end{align*}$
::9. 9. 3x2+6x+4

10. $\begin{align*}3y^2+9y+15\end{align*}$
::10. 3y2+9y+15

11. $\begin{align*}2x^2-12x+1\end{align*}$
::11. 2x2 - 12x+1

Complete the square for $\begin{align*}x\end{align*}$ and/or $\begin{align*}y\end{align*}$ in each of the following equations:
::下列方程式中的 x 和/或 Y , 填全正方形 :

12. $\begin{align*}4x^2-16x+y^2+2y=-1\end{align*}$
::12. 4x2 - 16x+y2+2y1

13. $\begin{align*}9x^2-54x+y^2-2y=-81\end{align*}$
::13. 9x2-54x+y2-2y_2y81

14. $\begin{align*}3x^2-6x-4y^2=9\end{align*}$
::14. 3x2-6x-4y2=9

15. $\begin{align*}y=x^2+4x+1\end{align*}$
::15.y=x2+4x+1

Review (Answers)
::回顾(答复)

Please see the Appendix.
::请参看附录。

Section outline

Introduction ::导言

Conics ::二次二次曲线

Standard F orm of a Conic ::二次曲线的标准窗体

Examples ::实例

Summary ::摘要

Review ::回顾

Review (Answers) ::回顾(答复)

Introduction
::导言

Conics
::二次二次曲线

Standard F orm of a Conic
::二次曲线的标准窗体

Examples
::实例

Summary
::摘要

Review
::回顾

Review (Answers)
::回顾(答复)