Section: 椭圆 | 9.4 椭圆 | The Way To Learn

Section outline

An ellipse is commonly known as an oval. Ellipses are just as common as in the real world with their own uses. Rooms that have elliptical shaped ceilings are called whisper 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“奥瓦尔 ” 。椭圆与实际世界一样常见,有其自身用途。有椭圆形天花板的房间被称为“耳语室 ” , 因为如果你站在一个焦点和耳语, 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 that are a set distance from two focus points while circles are defined as the collection of points that are a set distance from one center point. A third 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 how strangely shaped an ellipse is?
::椭圆看起来与圆形相似, 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点收集定点距离的点, 而圆则被定义为从一个中心点收集定点距离的点。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。一些椭圆是狭小的, 有些几乎是圆形的。您如何测量奇异地构造的椭圆形 ?

Graphing Ellipses
::绘制椭圆图

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 that there is some slack between them. Then use a pencil to pull the string taught 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.
::椭圆有两个方形。对于椭圆的每一个点, 与每个方形的距离总和是恒定的。这就是对椭圆的定义。另一种思考椭圆定义的方法是分配一组字符串, 并固定字符串的两端, 这样它们之间就会有一些松动。然后用铅笔拉动所教的字符串, 并沿着两个固定点追踪曲线。您将追踪椭圆, 字符串的固定端点将是 foci 。 Foci 是焦点的多元形式。在下面的图片中, (k) 是椭圆的中心, 而另外两个标记点是 foci 。

The major axis is the longest distance from end to end of an ellipse and is twice as long as the semi-major axis . The semi-major axis is the distance from the center of the ellipse to the furthest point on the ellipse and the semi-minor axis is the distance from the center to the edge of the ellipse on the axis that is perpendicular to the semi-major axis.
::主轴是从椭圆的尽端到尽端的最长距离,是半主轴的两倍。半主轴是从椭圆的中心到椭圆的最远点的距离,半主轴是半主轴与半主轴之间的中心到椭圆边缘的距离。

The general equation for an ellipse is:
::椭圆的一般等式是:

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

In this case the major axis is horizontal because $\begin{aligned} a \end{aligned}$ , the $\begin{aligned} x \end{aligned}$ -radius, is larger. If the $\begin{aligned} y \end{aligned}$ -radius were larger, then $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ would reverse. In other words, the coefficient $\begin{aligned} a \end{aligned}$ always comes from the length of the semi major axis (the longer one) and the coefficient $\begin{aligned} b \end{aligned}$ always comes from the length of the semi minor axis (the shorter one).
::在这种情况下,主轴是水平的,因为 a, x- radyus 较大。如果 y- radyus 更大, 那么则a 和 b 反转。换句话说, 系数a 总是来自半主轴的长度( 更长的轴) , 系数b 总是来自半次要轴的长度( 较短的轴 ) 。

In order to find the locations of the two foci, you will need to find the focal radius represented as $\begin{aligned} c \end{aligned}$ using the following relationship:
::为了找到两个角的位置,您需要使用以下关系找到以 c 表示的焦半径 :

$\begin{aligned} a^{2} - b^{2} = c^{2} \end{aligned}$
::a2-2-b2=c2

Once you have the focal radius, measure from 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 less circular. Eccentricity is calculated by:
::一旦您拥有焦半径, 则从中心处测量主轴以定位方形。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的数值则不那么圆。偏心度由以下因素计算:

$\begin{aligned} e = \frac{c}{a} \end{aligned}$
::电子=ca

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 $\begin{aligned} \frac{a^{2}}{c} \end{aligned}$ .
::椭圆也有两个直线直线线,与每个焦线相对应,但位于椭圆外部。从椭圆中心到每个直线直线的距离是a2c。

Examples
::实例

Example 1
::例1

Earlier, you were asked how you measure how strangely shaped an ellipse is. Ellipses are measured using their eccentricity. Here are three ellipses with estimated eccentricity for you to compare.
::早些时候, 有人问您如何测量椭圆的怪异形状。椭圆是用其偏心来测量的。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。

Example 2
::例2

Find the vertices (endpoints of the major axis), foci and eccentricity of the following ellipse.
::查找下列椭圆的顶部(主轴的端点)、角和偏心。

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

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 vertices are (2, 3) and (2, -5).
::椭圆的中心为(2, - 1),主轴为垂直,表示半主轴为a=4。顶部为(2, 3)和(2, 5)。

\begin{aligned} 16 - 4 & = c^{2} \\ 12 & = c \end{aligned}

::16-4=c212=c

Thus the foci are $\begin{aligned} (2, - 1 + \sqrt{12}) \end{aligned}$ and $\begin{aligned} (2, - 1 - \sqrt{12}) \end{aligned}$
::因此,本要素是(2,-1+12)和(2,-1-12)

Example 3
::例3

Sketch the following ellipse.
::绘制以下的椭圆。

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

Plotting the foci are usually important, but in this case the question simply asks you to sketch the ellipse. All you need is the center, $\begin{aligned} x \end{aligned}$ -radius and $\begin{aligned} y \end{aligned}$ -radius.
::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。您需要的是中心、 X 半径和 Y 半径。

Example 4
::例4

Put the following conic into graphing form.
::将以下二次曲线放入图形形式。

$\begin{aligned} 25 x^{2} - 150 x + 36 y^{2} + 72 y - 639 = 0 \end{aligned}$
::25x2 - 150x+36y2+72y- 639=0

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

::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 5
::例5

Put the following conic into graphing form.
::将以下二次曲线放入图形形式。

$\begin{aligned} 9 x^{2} - 9 x + 4 y^{2} + 12 y + \frac{9}{4} = - 8 \end{aligned}$
::9x2-9x+4y2+12y+94+8

\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(x2-122+4)(y2+3y+94)_8+4}949(x-12)2+4(y+32)2=1(x-12)2=1(x)219+(y+32)214=1

Summary

An ellipse is defined as the collection of points that are a set distance from two focus points, and has both an x-radius and a y-radius.
::椭圆的定义是收集从两个焦点点起定点距离的点数,这些点数既有x-半径,也有y-半径。

The major axis is the longest distance from end to end of an ellipse.
::主轴是从椭圆的尽头到最后的最长距离。

The semi-major axis is the distance from the center of the ellipse to the furthest point on the ellipse
::半主轴是从椭圆中心到椭圆最远点的距离

The semi-minor axis is the distance from the center to the edge of the ellipse on the axis perpendicular to the semi-major axis.
::半minor轴是半minor轴与半major轴垂直轴的椭圆中央至椭圆边缘之间的距离。

The general equation for an ellipse is: $\begin{aligned} \frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1, \end{aligned}$ where $\begin{aligned} a \end{aligned}$ is the semi major axis and $\begin{aligned} b \end{aligned}$ is the semi minor axis.
::椭圆的一般方程是x-h)2a2+(y-k)2b2=1,其中 a 是半主要轴, b 是半次要轴。

The focal radius $\begin{aligned} c \end{aligned}$ can be found using the relationship: $\begin{aligned} a^{2} - b^{2} = c^{2} . \end{aligned}$
::焦半径 c 可使用以下关系查找 : a2-b2=c2。

Eccentricity is the measure of how circular the shape is, calculated by: $\begin{aligned} e = \frac{c}{a} . \end{aligned}$
::偏心度是测量形状如何圆形的尺度,计算方法是:e=ca。

Ellipses also have two directrix lines that correspond to each focus but on the outside of the ellipse, with the distance from the center of the ellipse to each directrix line being $\begin{aligned} \frac{a^{2}}{c} . \end{aligned}$
::椭圆还有两条直径直径直线,与每个焦线相对应,但位于椭圆外部,从椭圆中心到每条直径直线的距离为a2c。

Review
::回顾

Find the vertices, foci, and eccentricity for each of the following ellipses.
::找到以下各省略的脊椎、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

Now sketch each of the following ellipses (note that they are the same as the ellipses in #1 - #3).
::现在绘制下方的省略号(请注意它们与 #1 - #3中的省略号相同)。

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.
::将以下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)的一个小轴线的椭圆和2、4的一个焦点(2、4)。

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)

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

Graphing Ellipses ::绘制椭圆图

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)