Course: 11.5 椭圆 | The Way To Learn

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椭圆
Introduction
::导言

An ellipse is commonly known as an oval. I n the real world, e llipses are just as common as , with their own uses. Rooms that have elliptical-shaped ceilings are called "whispering rooms," because if you stand at one focus point and whisper, someone standing at the other focus point will be able to hear you.
::椭圆通常被称为 oval 。在现实世界中, 椭圆和自己的用途一样常见。有椭圆形天花板的房间被称为“耳语室 ” , 因为如果你站在某个焦点并低声低语, 站在另一个焦点的人就能听到你的声音。

Ellipses look similar to circles, but there are a few key differences between these shapes. Ellipses have both an $\begin{aligned} x \end{aligned}$ -radius and a $\begin{aligned} y \end{aligned}$ -radius , while circles have only one radius. Another difference between circles and ellipses is that an ellipse is defined as the collection of points a set distance from two focal points, while circles are defined as the collection of points a set distance from one center point. A 3rd difference between ellipses and circles is that not all ellipses are similar to each other, while all circles are similar to each other. Some ellipses are narrow and some are almost circular. How do you measure and graph an ellipse?
::椭圆看上去与圆形相似, 但这些形状之间有一些关键区别。椭圆有X- 半径和 y- 半径, 而圆只有一个半径。圆和椭圆之间的另一个区别是, 椭圆被定义为从两个焦点收集点的定点距离, 而圆则被定义为从一个中心点收集点的定点距离。椭圆和圆之间的第三个区别是, 并非所有的椭圆都彼此相似, 而所有的圆都是相似的。有些椭圆是窄的, 有些几乎是圆的。您如何测量和绘制椭圆?

The Ellipse
::椭圆

Recall that an ellipse is generated when a plane is tilted so that it intersects both generator lines of one nappe of the cone. Thus, an ellipse has two foci . For every point on the ellipse, the sum of the distances to each foci is constant. This is what defines an ellipse. Another way of thinking about the definition of an ellipse is to allocate a set amount of string and fix the two ends of the string so there is some slack between them. Then use a pencil to pull the string taut and trace the curve all the way around both fixed points. You will trace an ellipse, and the fixed end points of the string will be the foci. (Foci is the plural form of focus.) In the picture below, $\begin{aligned} (h, k) \end{aligned}$ is the center of the ellipse, and the other two marked points are the foci.
::提醒注意, 当平面倾斜时, 椭圆会生成一个椭圆, 从而将锥形的一个环形的两个生成线相互交叉。因此, 椭圆会有两个 foci 。对于椭圆的每一个点, 与每个 foci 的距离总和是不变的。这是对椭圆的定义。有关椭圆定义的另一种思维方式是分配一组字符串, 并固定弦的两端, 从而在它们之间有一些松动。然后用铅笔拉起弦线, 并追踪两个固定点的曲线。您将追踪椭圆, 字符串的固定终点将是 foci 。 (Foci is the multic of focus.) 在下面的图片中, (h, k) 是椭圆的中心, 另外两个标记点是 foci 。

Standard Equation for an Ellipse
::椭圆的标准方程式

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

:x-h)2a2+(y-k)2b2=1(x-h)2b2+(y-k)2a2=1,

where $\begin{aligned} (h, k) \end{aligned}$ is the center of the circle , and $\begin{aligned} a > b \end{aligned}$ are the lengths of the semi-major and semi-minor axes.
::其中 (h, k) 是圆的中心, a>b 是半主轴和半小轴的长度。

In the example shown above, the major axis is the horizontal axis and t he minor axis is the vertical axis because $\begin{aligned} a > b \end{aligned}$ . In this case, the first standard equation for an ellipse would be used. If the $\begin{aligned} y \end{aligned}$ -radius were larger than the $\begin{aligned} x \end{aligned}$ -radius, the major axis would be the vertical axis and t he minor axis would be the horizontal axis. In this case, the second standard equation would be used. In general, the coefficient $\begin{aligned} a \end{aligned}$ always comes from the length of the semi-major axis (half of the longer axis), and the coefficient $\begin{aligned} b \end{aligned}$ always comes from the length of the semi-minor axis (half of the shorter axis).
::在上文所举的例子中,主要轴是水平轴,次要轴是垂直轴,因为 a>b。在这种情况下,将使用椭圆的第一个标准方程式。如果Y-弧度大于X-弧度,则主要轴是垂直轴,小轴是水平轴。在这种情况下,将使用第二个标准方程式。一般来说,一个系数总是来自半主轴的长度(长轴的一半),而系数b总是来自半偏角轴的长度(短轴的一半)。

To find the locations of the two foci, you will need to find the focal length represented as $\begin{aligned} c \end{aligned}$ using the following relationship: $\begin{aligned} a^{2} - b^{2} = c^{2} \end{aligned}$ .
::要找到两个角的位置,您需要使用以下关系来找到以 c 表示的焦距: a2-b2=c2。

Add the focal length to the corresponding coordinate of the center along the major axis to locate the foci. The general shape of an ellipse is measured using eccentricity. Eccentricity is a measure of how oval or how circular the shape is. Ellipses can have an eccentricity between 0 and 1, where a number close to 0 is extremely circular, and a number close to 1 is more elongated or flatter. Eccentricity is calculated by $\begin{aligned} e = \frac{c}{a} \end{aligned}$ .
::按主轴向中心相应的坐标添加焦长以定位方形。对椭圆的一般形状使用偏心度来测量。偏心度是测量奥瓦尔或圆形的尺度。椭圆的偏心度介于0到1之间,其中接近0的数值极圆,接近1的数值较长或偏重。偏心度由 e=ca 计算。

Eccentricity
::偏心

$\begin{aligned} e = \frac{c}{a}, \end{aligned}$ where $\begin{aligned} a \end{aligned}$ is the length of the semi-major axis and $\begin{aligned} c = \sqrt{a^{2} - b^{2}} \end{aligned}$ .
::e=ca,其中 a 是半主轴的长度, c=a2-b2。

Ellipses also have two directrix lines that correspond to each focus, but on the outside of the ellipse. The distance from the center of the ellipse to each directrix line is $\begin{aligned} x = \frac{a^{2}}{c} \end{aligned}$ .
::椭圆也有两个直线直线线,与每个焦线相对应,但在椭圆外部。从椭圆中心到每个直线直线的距离是 x=a2c。

D irectrix line
::直导线

$\begin{aligned} x = \frac{a^{2}}{c}, \end{aligned}$ where $\begin{aligned} a \end{aligned}$ is the length of the semi-major axis and $\begin{aligned} c = \sqrt{a^{2} - b^{2}} \end{aligned}$ .

The following video defines an ellipse and explains how to graph an ellipse in standard form:
::以下影片定义了椭圆,并解释了如何以标准格式绘制椭圆图:

The following video explains how to graph an ellipse in general form:
::以下影片解释如何以一般形式绘制椭圆图:

Play, Learn, and Explore Ellipses: .
::播放、学习和探索椭圆:.

Examples
::实例

Example 1
::例1

Find the endpoints of the major axis, foci, and eccentricity of the following ellipse:
::查找下列椭圆的主要轴、方形和偏心的终点:

$\begin{aligned} \frac{x^{2}}{25} + \frac{y^{2}}{16} = 1. \end{aligned}$

::x225+y216=1

Solution:
::解决方案 :

The center of this ellipse is at (0, 0). The semi-major axis is horizontal with $\begin{aligned} a = 5 \end{aligned}$ . This means that the endpoints are at (5, 0) and (-5, 0). The semi-minor axis is vertical with $\begin{aligned} b = 4 \end{aligned}$ .
::此椭圆的中心值为 0, 0 。半主轴水平为 a=5 。这意味着端点为 (5, 0) 和 (5, 0) 。半最小轴为 b=4 垂直。

$\begin{aligned} 25 - 16 & = c^{2} \\ 3 & = c \end{aligned}$

::25-16=c23=c

The focal radius is 3. This means that the foci are at (3, 0) and (-3, 0).
::焦半径为 3 。这意味着角在( 3, 0) 和( 3, 0) 。

The eccentricity is $\begin{aligned} e = \frac{3}{5} \end{aligned}$ .
::偏心度是e=35

Example 2
::例2

Sketch the following ellipse:
::绘制下列椭圆:

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

:y-1)216+(x-2)29=1。

Solution:
::解决方案 :

Plotting the foci is usually important, but in this case the question simply asks you to sketch the ellipse. All you need are the center and the lengths of the semi-major and semi-minor axes . The center is $\begin{aligned} (2, 1) \end{aligned}$ with $\begin{aligned} a = 4 \end{aligned}$ and $\begin{aligned} b = 3 \end{aligned}$ .
::绘制方形图通常很重要, 但在此情况下, 问题只是要求您绘制椭圆。您需要的只是半主轴和半小轴的中心和长度。中心为 2, 1 和 a=4 和 b=3 。中心为 2, 1 和 a=4 和 b= 3 。

Example 3
::例3

Put the following conic into graphing form:
::将下列二次曲线放入图形形状:

$\begin{aligned} 25 x^{2} - 150 x + 36 y^{2} + 72 y - 639 = 0. \end{aligned}$

::25x2-150x+36y2+72y-639=0。

Solution:
::解决方案 :

Complete the square to put in graphing form.
::完成方形后, 将方形放入图形格式。

$\begin{aligned} 25 x^{2} - 150 x + 36 y^{2} + 72 y - 639 & = 0 \\ 25 (x^{2} - 6 x) + 36 (y^{2} + 2 y) & = 639 \\ 25 (x^{2} - 6 x + 9) + 36 (y^{2} + 2 y + 1) & = 639 + 225 + 36 \\ 25 (x - 3)^{2} + 36 (y + 1)^{2} & = 900 \\ \frac{25 (x - 3)^{2}}{900} + \frac{36 (y + 1)^{2}}{900} & = \frac{900}{900} \\ \frac{(x - 3)^{2}}{36} + \frac{(y + 1)^{2}}{25} = 1 \end{aligned}$

::25x2 - 150x+36y2+72y+639=025(x2- 6x)+36(y2+2y)=63925(x2- 6x+9)+36(y2+2y+1)=639+225+3625(x3)2+36(y+1)2=900025(x-33)2900+36(y+1)290900900(x-33)2336+(y+1)225=1

Example 4
::例4

Recall the problem from the Introduction: H ow do you measure and graph an ellipse?
::回顾导言中的问题:如何测量和绘制椭圆?

Solution:
::解决方案 :

Ellipses are measured using their eccentricity. Here are three ellipses with estimated eccentricity for you to compare.
::椭圆是用它们的偏心来测量的。这里有三个具有估计偏心的椭圆,可供您比较。

Eccentricity is the ratio of the focal radius to the semi-major axis: $\begin{aligned} e = \frac{c}{a} \end{aligned}$ .
::偏心度是焦半径与半主轴(e=ca)之比。

You can graph ellipses using the foci, axes, and center.
::您可以使用方形、轴和中枢绘制椭圆图。

Example 5
::例5

Find the endpoints of the major axis, foci, and eccentricity of the following ellipse:
::查找下列椭圆的主要轴、方形和偏心的终点:

$\begin{aligned} \frac{(x - 2)^{2}}{4} + \frac{(y + 1)^{2}}{16} = 1. \end{aligned}$

:x-2)24+(y+1)216=1。

Solution:
::解决方案 :

The center of the ellipse is at (2, -1). The major axis is vertical, which means the semi-major axis is $\begin{aligned} a = 4 \end{aligned}$ . The endpoints are (2, 3) and (2, -5).
::椭圆的中心是(2, - 1),主轴是垂直的,这意味着半主轴是a=4。终点是(2, 3)和(2, 5)。

$\begin{aligned} 16^{2} - 4^{2} & = c^{2} \\ 4 \sqrt{15} = \sqrt{240} & = c \end{aligned}$

::162-42=c2415=240=c

Thus, the foci are $\begin{aligned} (2, - 1 + 4 \sqrt{15}) \end{aligned}$ and $\begin{aligned} (2, - 1 - 4 \sqrt{15}) . \end{aligned}$
::因此,核心是(2,-1+415)和(2,-1-415)。

The eccentricity is $\begin{aligned} e = \frac{c}{a} = \frac{4 \sqrt{15}}{4} = \sqrt{15} \end{aligned}$ .
::偏心度是e=ca=4154=15。

Example 6
::例6

Sketch the following ellipse:
::绘制下列椭圆:

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

:x-3)2+(y-1)29=1。

Solution:
::解决方案 :

Example 7
::例7

Convert the following conic into graphing form:
::将以下二次曲线转换为图形形状:

$\begin{aligned} 9 x^{2} - 9 x + 4 y^{2} + 12 y + \frac{9}{4} = - 8. \end{aligned}$

::9x2-9x+4y2+12y+948。

Solution:
::解决方案 :

Complete the square to convert.
::完成正方形转换。

$\begin{aligned} 9 x^{2} - 9 x + 4 y^{2} + 12 y + \frac{9}{4} & = - 8 \\ 9 x^{2} - 9 x + \frac{9}{4} + 4 y^{2} + 12 y & = - 8 \\ 9 (x^{2} - x + \frac{1}{4}) + 4 (y^{2} + 3 y) & = - 8 \\ 9 {(x - \frac{1}{2})}^{2} + 4 (y^{2} + 3 y + \frac{9}{4}) & = - 8 + 4 \cdot \frac{9}{4} \\ 9 {(x - \frac{1}{2})}^{2} + 4 {(y + \frac{3}{2})}^{2} & = 1 \\ \frac{{(x - \frac{1}{2})}^{2}}{\frac{1}{9}} + \frac{{(y + \frac{3}{2})}^{2}}{\frac{1}{4}} = 1 \end{aligned}$

::9x2- 9x+4y2+12y+9489x2-9x+94+4y2+12y_89(x2-x+14+4(y2+3y)\89(x-12+3y)\89(y2+12)2+4(y2+3y+94)_8+4}949(x-12)2+4(y+322)2=1(x-12)2=219+(y+32)214=1

Summary
::摘要

An ellipse is the collection of points whose sum of distances from two foci is constant.
::椭圆是指从两个方位之间的距离总和保持不变的点的集合点。

The foci in an ellipse are the two points that the ellipse curves around.
::椭圆的方形是椭圆曲线的两点。

Eccentricity is a measure of how oval or how circular the shape is. It is the ratio of the focal radius to the semi-major axis: $\begin{aligned} e = \frac{c}{a} \end{aligned}$ .
::偏心度是测量形状的奥瓦尔或圆形的尺度。它是焦半径与半主轴(e=ca)的比例。

The major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter.
::椭圆的主要轴是其最长的直径:一个穿过中间和角的线段,以周界最宽的点为终点。

The semi-major axis is one half of the major axis. It runs from the center, through a focus, and to the endpoints . It is the longest radius of an ellipse.
::半主轴是主轴的半径。它从中心、通过焦点和终点运行。它是椭圆的最长半径。

The semi-minor axis is the shortest radius of an ellipse.
::半最小轴是椭圆的半径。

Review
::回顾

Find the vertices, foci, and eccentricity for each of the following ellipses:
::找到以下每个省略号的脊椎、角和偏心度:

1. $\begin{aligned} \frac{(x - 1)^{2}}{4} + \frac{(y + 5)^{2}}{16} = 1 \end{aligned}$
::1. (x-1)24+(y+5)216=1

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

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

Sketch each of the ellipses below. (Note they are the same as the ellipses in 1-3 above.)
:注意它们与上面1-3的省略号相同。 )

4. $\begin{aligned} \frac{(x - 1)^{2}}{4} + \frac{(y + 5)^{2}}{16} = 1 \end{aligned}$
::4. (x-1)24+(y+5)216=1

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

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

Put each of the following equations into graphing form:
::将下列方程式中的每一个方程式都放入图形表格:

7. $\begin{aligned} x^{2} + 2 x + 4 y^{2} + 56 y + 197 = 16 \end{aligned}$
::7. x2+2x+4y2+56y+197=16

8. $\begin{aligned} x^{2} - 8 x + 9 y^{2} + 18 y + 25 = 9 \end{aligned}$
::8. x2-8x+9y2+18y+25=9

9. $\begin{aligned} 9 x^{2} - 36 x + 4 y^{2} + 16 y + 52 = 36 \end{aligned}$
::9. 9x2-36x+4y2+16y+52=36

Find the equation for each ellipse based on the description:
::根据描述查找每个椭圆的方程 :

10. An ellipse with vertices (4, -2) and (4, 8), and minor axis of length 6.
::10. 有脊椎的椭圆(4,-2)和(4,8)及6号小轴的椭圆。

11. An ellipse with minor axis from (4, -1) to (4, 3), and major axis of length 12.
::11. 具有小轴从(4,-1)至(4,3)和大轴为12的椭圆。

12. An ellipse with minor axis from (-2, 1) to (-2, 7), and one focus at (2, 4).
::12. 从(-2)至(-2)至(-2)至(7)以小轴为轴的椭圆,一个焦点(2,4)为轴。

13. An ellipse with one vertex at (6, -15), and foci at (6, 10) and (6, -14).
::13. 在(6,15)和(6,10)和(6,14)之间有一个顶部的椭圆形和角形。

A bridge over a roadway is to be built with its bottom the shape of a semi-ellipse 100 feet wide and 25 feet high at the center. The roadway is to be 70 feet wide.
::要在公路上建造一座桥,其底部是100英尺宽、25英尺高的半椭圆形,中间是70英尺宽。

14. Find one possible equation of the ellipse that models the bottom of the bridge.
::14. 寻找一个可能的等离子体方程式,用来模拟桥底。

15. What is the clearance between the roadway and the overpass at the edge of the roadway?
::15. 公路与公路边缘的过桥之间的清关是什么?

Review (Answers )
::回顾(答复)

Please see the Appendix.
::请参看附录。

Section outline

General

椭圆

Introduction ::导言

The Ellipse ::椭圆

Standard Equation for an Ellipse ::椭圆的标准方程式

Eccentricity ::偏心

D irectrix line ::直导线

Examples ::实例

Summary ::摘要

Review ::回顾

Review (Answers ) ::回顾(答复)

Introduction
::导言

The Ellipse
::椭圆

Standard Equation for an Ellipse
::椭圆的标准方程式

Eccentricity
::偏心

D irectrix line
::直导线

Examples
::实例

Summary
::摘要

Review
::回顾

Review (Answers )
::回顾(答复)