12.11 摘要:锥体部分

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• Select section 摘要: 二次曲线区块

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  摘要: 二次曲线区块

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  In this chapter, we learned about: 
  ::在本章中,我们了解到:

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  Distance on the Number Line
  ::数字行的距离

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    The distance, d , between two points in the coordinate plane, $\begin{array}{r}(x_1,y_1)\end{array}$  and $\begin{array}{r}(x_2,y_2)\end{array}$ , can be found using the formula $\begin{array}{r}d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.\end{array}$  
    ::坐标平面中两个点(x1,y1)和(x2,y2)之间的距离(d),可用公式d=(x2-x1)2+(y2-y1)2找到。

    
    ::坐标平面中两个点(x1,y1)和(x2,y2)之间的距离(d),可用公式d=(x2-x1)2+(y2-y1)2找到。

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  The Conic Sections 
  ::二次曲线区域

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  • The conic sections are formed from the intersection of the surface of a double right-circular cone and a plane.
    ::这些二次曲线系从双右圆锥形和平面表面的交叉点形成。
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  • There are four conic sections—parabola, circle, ellipse, and hyperbola—and three degenerate forms—point, line, and two intersecting lines.
    ::有四个二次曲线段——帕拉波拉、圆圈、椭圆和双曲线——以及三个退化的形态点、线和两个交叉线。
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  • The general form of the equation of a conic section is $\begin{array}{r}A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0,\end{array}$   where $\begin{array}{r}A,B,C,D,E,\end{array}$  and $\begin{array}{r}F\end{array}$  are real numbers.
    ::二次曲线段方程式的一般形式是 Ax2+Bxy+Cy2+Dx+Ey+F=0,其中A、B、C、D、E和F为实际数字。
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  • We can recognize what type of conic section the equation represents by comparing $\begin{array}{r}A\end{array}$  and $\begin{array}{r}C\end{array}$ . (In this chapter, we assume $\begin{array}{r}B=0\end{array}$ .) 
    ::我们可以通过比较A和C来确认方程代表的二次曲线部分的种类(在本章中,我们假定B=0)。

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  The Geometry of Conic Sections
  ::二次曲线段的几何

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  • A parabola is a curve such that any point on the curve is equidistant from another point, called the focus, and a line called the directrix.
    ::抛物线是一个曲线,因此曲线上的任何点从另一个点(称为焦点)和直线(称为直线)的位置相等。
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  • A circle is a set of points that are equidistant from a center point.  
    ::a 圆是一组从中点开始等距的点。
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  • An ellipse is the set of all points such that the sum of the distances from two fixed points, called foci (the plural of focus), is constant.
    ::椭圆是所有点的组合,使两个固定点(称为焦点复数)的距离总和保持不变。
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  • A hyperbola is the set of all points such that the differences of the distances from the foci are constant.
    ::双倍波拉是所有点的组合, 以便与方位距离的距离差异是恒定的 。

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  Equations of Conic Sections
  ::二次曲线各部分的平方

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  • To find an equation of a conic section in standard form, we complete the square.
    ::为了在标准格式中找到二次曲线的方程式,我们完成方形。
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  • The standard forms of the equation of a parabola are  $\begin{array}{r}(x-h{)}^2=4p(y-k)\end{array}$  (vertical axis of symmetry) and  $\begin{array}{r}(y-k{)}^2=4p(x-h)\end{array}$  (horizontal axis of symmetry).
    ::抛物线方程的标准形式是(x-h)2=4p(y-k)(对称的垂直轴)和(y-k)2=4p(x-h)(对称的横向轴)。
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  • The standard form of an equation of a circle is $\begin{array}{r}(x-h{)}^2+(y-k{)}^2=r^2,\end{array}$   where $\begin{array}{r}(h,k)\end{array}$   are the coordinates of the center, and $\begin{array}{r}r\end{array}$  is the radius.
    ::圆形方程式的标准形式是 (x-h)2+(y-k)2=r2, 其中 (h,k) 是中心坐标, r 是半径 。
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  • The standard forms of an equation of an ellipse are $\begin{array}{r}\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1\end{array}$   (horizontal major axis) and $\begin{array}{r}\frac{(x-h{)}^2}{b^2}+\frac{(y-k{)}^2}{a^2}=1\end{array}$   (vertical major axis). We assume $\begin{array}{r}a>b\end{array}$ . 
    ::椭圆方程式的标准形式是 (x-h) 2a2+(y-k) 2b2=1 (横向主要轴) 和 (x-h) 2b2+(y-k) 2a2=1 (垂直主轴)。
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  • The standard forms of an equation of a hyperbola are $\begin{array}{r}\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1\end{array}$   (horizontal transverse axis ) and $\begin{array}{r}\frac{(y-k{)}^2}{a^2}-\frac{(x-h{)}^2}{b^2}=1\end{array}$   (vertical transverse axis ). 
    ::双倍波拉方程式的标准形式为(x-h)2a2-(y-k)2b2=1(横向横轴)和(y-k)2a2-(x-h)2b2=1(垂直横轴)。

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  Graphing Conic Sections
  ::绘制二次曲线区域图

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  • To graph a parabola, find the vertex and $\begin{array}{r}p\end{array}$ . Identify the focus and then count $\begin{array}{r}2p\end{array}$  units away from the focus perpendicular to the axis of symmetry. Draw a parabola through the vertex and these two points.  
    ::要绘制抛物线图,请找到顶点和 p. 标明焦点,然后从焦点垂直到对称轴的 2p 单位数。 通过顶点和这两个点绘制一个抛物线。
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  • To graph a circle, find the center and then count the radius number of units to the right, left, up, and down to get four points on the graph of the circle. Draw a circle to connect them.
    ::要绘制圆形图,请在圆形图上绘制圆形图,找到圆形中心,然后向右、向左、向上和向下计算单位的半径数,以获得圆形图上的四个点。绘制一个圆形以连接它们。
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  • To graph an ellipse, we find the center, the vertices, and the co-vertices. The ellipse goes through the vertices and co-vertices.
    ::要绘制椭圆图,我们找到中心, 脊椎, 和共圆。 椭圆穿过顶和共圆。
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  • To graph a hyperbola, draw a box using the center, a, and b. Then draw the asymptotes and the vertices. Last, graph the branches of the hyperbola. 
    ::要绘制双曲线图,请用中心( a) 和( b) 绘制一个框。然后绘制小行星和脊椎。最后,请绘制双曲线的分支。

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  Solving Systems of Conic Sections
  ::二次元件的溶解系统

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  • We can solve systems of conic sections by graphing, substitution, and elimination by addition.
    ::我们可以通过图形绘制、替代和添加消除等方法解决二次曲线部分的系统。

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  Looking Back, Looking Forward
  ::回顾,展望未来

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  In this chapter, we have considered conic sections and applications where the intersection of the surface of a double right-circular cone and a plane are useful. We have seen that we can use conic sections to locate lightning strikes, determine the path of a comet, and learn how to manage a fishery. 
  ::在本章中,我们考虑了双右圆锥形和飞机表面交叉点有用的二次曲线部分和应用。 我们已经看到,我们可以使用二次曲线部分来确定闪电打击的位置,确定彗星的路径,并学习如何管理渔业。

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  In the next chapter, we will consider sequences and series. Sequences and series define patterns, which, in some way, have been components of every chapter of this book, and are also part of our day-to-day lives.
  ::在下一章,我们将考虑顺序和系列。序列和系列定义模式,在某种程度上,这些模式是本书每一章的组成部分,也是我们日常生活的一部分。

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  Chapter Review
  ::回顾章次审查