Course: 10.1 锥体结块-interactive

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二次曲线区块

The conic sections are the shapes that can be created when a plane intersects a double-napped cone. In other words, the conic sections are the cross sections of a double-napped cone. Depending on the angle of the plane with respect to the cone, a conic section may be a circle , an ellipse , a parabola , or a hyperbola .
::二次曲线区域是当一平面交叉一个双层锥体时可以生成的形状。换句话说, 二次曲线区域是双层锥体的交叉部分。视平面相对于锥体的角度而定, 二次曲线区域可以是圆形、椭圆、 parbola 或双层。

These conic sections are shown in the interactive below with their general equations. In geometry, you will focus on circles and . You will study and in more depth in future courses.
::这些二次曲线部分在下文互动部分中显示,与它们的一般方程相同。在几何学中,您将关注圆圈和.您将在今后的课程中学习,并将更深入地学习。

INTERACTIVE

Conic Sections

In the interactive below, explore the conic sections created by the intersection of a plane and a double cone.
::在下文互动部分中,探索由平面和双锥体交点产生的二次曲线。

Click the start/stop button to rotate the plane through the double cone.
::单击起始/停止按钮通过双锥体旋转平面。

Observe what conic sections are created at each angle of rotation.
::观察每个旋转角度所创建的二次曲线区域。

Circle
::圆圆

A circle is the set of all points that are equidistant from a fixed point in the plane. A circle is obtained when the cutting plane is parallel to the base of a right cone.
::a 圆是从平面固定点上等距所有点的一组。当切开平面与右锥体底部平行时,将获得一个圆。

The standard equation of the circle centered at $\begin{aligned} (h, k) \end{aligned}$ , whose radius is $\begin{aligned} r \end{aligned}$ is
::圆以 (h,k) 为中心,圆的半径为 r 的圆圆的标准方程

$\begin{aligned} (x - h)^{2} + (y - k)^{2} = r^{2} \end{aligned}$
:x-h)2+(y-k)2=r2

Diagram illustrates a circle with center (h,k), radius r, and coordinate axes.

In a circle equation, $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ are both squared and the coefficients on them are the same, including the sign.
::在圆形方程中, x 和 y 均方形,其系数相同,包括符号。

Parabola
::帕拉波拉

A parabola is the set of points in a plane that are equidistant from both the directrix (a fixed straight line) and the focus (a fixed point). The parabola is obtained when the cutting plane is parallel to the lateral surface of the cone and passes through just one cone.
::抛物线是直线(固定直线)和焦点(固定点)等距的平面上的一组点数,当切割平面与锥体平面平行,通过一个锥体时,即可获得抛物线。

The standard equation of the parabola (axis of symmetry parallel to the $\begin{aligned} y \end{aligned}$ -axis) with vertex at $\begin{aligned} (h, k) \end{aligned}$ , the focus at $\begin{aligned} (h, k + p) \end{aligned}$ and the directrix $\begin{aligned} y = k - p \end{aligned}$ is
::以(h,k,k)点为顶点、以(h,k+p)点为焦点和直方向为y=k-p的抛星(与 Y 轴平行的对称轴)的标准方程

$\begin{aligned} (x - h)^{2} = 4 p (y - k) \end{aligned}$
:x-h)2=4p(y-k)

$\begin{aligned} p \end{aligned}$ is the distance from the vertex to the focus.
::p 表示从顶部到焦点的距离。

The vertex form of the equation, where $\begin{aligned} a = \frac{1}{4 p} \end{aligned}$ , is
::等式的顶部形式, a=14p 是

$\begin{aligned} y = a (x - h)^{2} + k \end{aligned}$
::y=a(x-h)2+k

In a parabola equation, either $\begin{aligned} x \end{aligned}$ or $\begin{aligned} y \end{aligned}$ is squared and not both. If the parabola contains $\begin{aligned} \pm x^{2} \end{aligned}$ , the graph opens in a $\begin{aligned} y \end{aligned}$ -direction (vertically up or down). If the parabola contains $\begin{aligned} \pm y^{2} \end{aligned}$ , the graph opens in an $\begin{aligned} x \end{aligned}$ -direction (horizontally left or right).
::在抛物线方程式中, x 或 y 是平方, 而不是两者兼有。如果抛物线含有 +x2, 图形以 y 方向打开( 垂直向上或向下) 。如果抛物线包含 +Y2, 图形以 x 方向打开( 向左或向右) 。

The equations of parabolas with vertex $\begin{aligned} (0, 0) \end{aligned}$ , in different orientations are as follows:
::具有顶点(0,0)的parabolas的方程以不同方向如下:

Graph of a parabola showing its focus and equation y² = 4ax.

Graph of a parabola with focus F and equation y² = -4ax.

Graph showing a parabola with vertex at the origin and focus marked at (0,a).

Graph of a parabola with vertex and focus indicated, illustrating its standard equation.

Ellipse
::椭圆

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. If the plane cuts entirely across one nappe of the cone and at an angle between $\begin{aligned} 0^{\circ} \end{aligned}$ and $\begin{aligned} 90^{\circ} \end{aligned}$ , then the curve of intersection of cone and plane is an ellipse.
::椭圆是飞机上所有点的组合,其距离与飞机上两个固定点之间的距离是固定的。如果飞机完全切开锥形的一个环形,以0和90之间的角切开,锥形与平面的交叉曲线就是一个椭圆。

The standard centered at $\begin{aligned} (h, k) \end{aligned}$ , whose horizontal radius is $\begin{aligned} a \end{aligned}$ and vertical radius is $\begin{aligned} b \end{aligned}$ is
::以 (h,k) 为标准中心,其水平半径为(h,k) ,垂直半径为 b。

$\begin{aligned} \frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1 \end{aligned}$
:x-h)2a2+(y-k)2b2=1

Graph of an ellipse labeled with its center, axes, and key points.

In an ellipse equation, $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ are both squared and the coefficients are positive but different.
::在椭圆方程式中,x和y是平方的,系数是正数,但不同。

Hyperbola
::超重波

When the cutting plane is at an angle to the base such that it passes through both the cones, the curve of intersection is a hyperbola. A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.
::当切割平面在向基底的一个角度,通过两个锥体时,十字路口的曲线是双波拉。双波拉是一个平面所有点的组合,其距离与平面两个固定点的距离是恒定的。

The standard equation of a hyperbola centered at $\begin{aligned} (h, k) \end{aligned}$ , whose horizontal radius is $\begin{aligned} a \end{aligned}$ and vertical radius is $\begin{aligned} b \end{aligned}$ is
::以 (h,k) 为中心的双倍波拉的标准方程式,其水平半径为(h,k),垂直半径为b

$\begin{aligned} \frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1 \end{aligned}$
:x-h)2a2-(y-k)2b2=1

Diagram illustrating the structure and properties of a hyperbola with key points labeled.

In a hyperbola equation, $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ are both squared, and exactly one of the coefficients are negative and exactly one of the coefficients are positive.
::在超重波拉方程式中,x和y都是正方形,确切地说,一个系数为负数,一个系数为正数。

INTERACTIVE

Types of Conic Sections

In the interactive, click on the arrow buttons to scroll through each of the four primary conics.
::在互动中,单击箭头按钮滚动通过四个主二次曲线。

Let's look at some problems involving conic sections.
::让我们来研究一下涉及二次曲线的一些问题

1. How is a circle created as the intersection of a double cone and a plane?
::1. 一个圆圈是如何作为双锥形和平面的交叉点形成的?

The cross section is taken parallel to the base of the cones, as shown below.
::截面与锥体的底部平行,如下所示。

2. How is an ellipse created as the intersection of a double cone and a plane?
::2. 如何作为双锥体和平面的交叉点而形成椭圆形?

The cross section is taken at a slight angle to the base of the cones, as shown below.
::如下文所示,截截部分以微小角度拍摄至锥锥体底部。

Examples
::实例实例实例实例

Example 1
::例1

Conic sections are a family of graphs that include circles and parabolas. Why are they called conic sections?
::二次曲线系由包含圆形和parapolas的图表组成的一组。为什么它们被称为二次曲线系呢?

Conic sections are called conic sections because they are the cross sections of a double cone . The word conic comes from the word cone, and the word section comes from cross section.
::二次曲线区段被称为二次曲线区段, 因为它们是双二次曲线的交叉区段。二次曲线区段来自二次曲线区段, 单词区段来自二次曲线区段。

Example 2
::例2

How is a parabola created as the intersection of a double cone and a plane?
::抛物线是如何作为双锥形和平面的交叉点而创造的?

The cross section is taken such that the intersection plane has the same slope as the lateral surface of the cone and passes through just one cone.
::十字路口的截面与锥形平面的斜坡相同,仅通过一个锥形。

Example 3
::例3

How is a hyperbola created as the intersection of a double cone and a plane?
::双层锥形和平面的交汇点如何创造出双层圆柱形和双层圆锥形的交汇点?

The cross section is taken at an angle to the base such that it passes through both the cones, as shown below.
::如下文所示,从一个角度将横截段取向基底,以便通过锥体和锥体。

Example 4
::例4

Are there any other cross sections of a double cone besides the four primary conic sections?
::除了四个主要的二次曲线外,是否还有其他双锥体交叉部分?

There are three other cross sections -- a point , a line , and a pair of intersecting lines . These are referred to as the , because they are not as complex. A degenerate conic is generated when a plane intersects the vertex of the cone. The degenerate form of a circle or an ellipse is a singular point. The degenerate form of a parabola is a line. The degenerate form of a hyperbola is two intersecting lines. The study of conic sections focuses on the four primary conic sections and not on the degenerate conics.
::有另外三个交叉区域 -- -- 一个点、一条线和一对交叉线。这些被称为交叉线,因为它们不那么复杂。当平面交叉锥体的顶部时,就会产生退化的二次曲线。圆或椭圆的退化形式是一个单点。抛物线的退化形式是一条线。超博拉的退化形式是两条交叉线。对二次曲线部分的研究侧重于四个主要的二次曲线部分,而不是退化的二次曲线。

Summary

The conic sections are the shapes that can be created when a plane intersects a double-napped cone.
::二次曲线区域是当平面交叉一个双层锥体时可以生成的形状。

A cross section is the intersection of a three-dimensional solid with a plane.
::横截面是三维固体与平面的交叉点。

A circle is a set of all points that are the same distance away from a specific point, called the center. The standard form of a circle with center $\begin{aligned} (h, k) : \end{aligned}$

$\begin{aligned} (x - h)^{2} + (y - k)^{2} = r^{2} \end{aligned}$

::a 圆是一组所有点的集合点,这些点与特定点的距离相同,称为中点。圆的标准形式与中(h,k): (x-h)2+(y-k)2=r2

A parabola is the set of points in a plane that are equidistant from both the directrix (a fixed straight line) and the focus (a fixed point). The standard form of a parabola with center $\begin{aligned} (h, k) : \end{aligned}$
$\begin{aligned} (x - h)^{2} = 4 p (y - k) \end{aligned}$

::抛物线是指从直线(固定直线)和焦点(固定点)之间等距的平面上的一组点。抛物线是中(h,k)x-h)2=4p(y-k)的抛物线的标准形式。

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. The standard form of an ellipse with center $\begin{aligned} (h, k) : \end{aligned}$
$\begin{aligned} \frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1 \end{aligned}$

::椭圆是一个平面中所有点的集合,其距离与平面中两个固定点的距离之和是一个常数。正中(h,k): (x-h)2a2+(y-k)2b2=1

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant. The standard form of a hyperbola with center $\begin{aligned} (h, k) : \end{aligned}$
$\begin{aligned} \frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1 \end{aligned}$

::Hyperbola 是一个平面所有点的组合, 平面上两个固定点的距离是恒定的差值。以 (h, k) (x- h) 2a2 - (y- k) 2b2=1 为中位的超大波拉的标准格式。

Review
::审查审查审查审查

1. What are the four primary conic sections?
::1. 四个主要二次曲线部分是什么?

2. Why are the conic sections called conic sections ?
::2. 为什么二次曲线部分被称为二次曲线部分?

Name each of the conic sections based on its picture or equation.
::根据其图片或方程列出每个二次曲线区域的名称。

3. $\begin{aligned} (x - 2)^{2} + (y + 5)^{2} = 1 \end{aligned}$
::3. (x-2)2+(y+5)2=1

5. $\begin{aligned} \frac{(x - 2)^{2}}{4} + \frac{(y + 5)^{2}}{9} = 1 \end{aligned}$
::5. (x-2)24+(y+5)29=1

6. $\begin{aligned} - \frac{(x - 2)^{2}}{4} + \frac{(y + 5)^{2}}{9} = 1 \end{aligned}$
::6.-(x-2)24+(y+5)29=1

8. $\begin{aligned} (x - 2)^{2} = 4 (y - 5) \end{aligned}$
::8. (x-2-2)2=4(y-5)

10. $\begin{aligned} \frac{(y + 2)^{2}}{4} + \frac{(x + 5)^{2}}{9} = 1 \end{aligned}$
::10. (y+2)24+(x+5)29=1

11.

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

13. $\begin{aligned} x = \frac{1}{4} (y - 1)^{2} + 5 \end{aligned}$
::13. x=14(y- 1)2+5

14. What are the degenerate conics?
::14. 什么是堕落的二次曲线?

15. Why are the degenerate conics called degenerate conics?
::15. 为什么堕落的二次曲线被称为二次曲线?

16. A semielliptical archway over a one-way road has a height of 10 feet and a width of 40 feet. You are driving in a truck that is 10 feet wide and 9 feet high. Will your truck make it under the archway? Justify your answer.
::16. 单行道上的半电子拱门高度为10英尺,宽度为40英尺,驾驶卡车的宽度为10英尺,高度为9英尺,卡车能否在拱门下行驶,请说明答案。

17. The sound of an explosion is picked up by two microphones that are two miles apart. Microphone A picked up the sound 4 seconds before microphone B. Assuming the sound travels at a speed of 1,100 feet per second, determine the possible locations of the explosion with respect to the locations of the microphones.
::17. 爆炸声由相隔两英里的两部麦克风接收,麦克风A在麦克风B前4秒钟接收到音响。假定音速以每秒1 100英尺的速度移动,确定麦克风位置的可能爆炸地点。

Review (Answers)
::审查(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

General

二次曲线区块

Circle ::圆圆

Parabola ::帕拉波拉

Ellipse ::椭圆

Hyperbola ::超重波

Examples ::实例实例实例实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::审查审查审查审查

Review (Answers) ::审查(答复)