摘要:二次曲线

In this chapter, you  learned that a conic section is the family of shapes that are formed by the different ways a flat plane intersects a two-sided cone in three-dimensional space. Parabolas, circles, ellipses, and hyperbolas each have precise definitions that are important to their shapes, as well as equations that can be written in a form that provides key information for sketching and graphing. Each conic has important points and measurements. Relationships between these key numbers are used to understand how the conic is formed, create equations, and sketch the graphs of the equations.
::在本章中,您了解到一个二次曲线部分是由平面在三维空间中以不同方式交叉两面锥体形成的形状组成的组合。 帕拉波拉斯、圆圈、椭圆和超双形各有对其形状很重要的精确定义,以及可以以为草图和图形提供关键信息的形式写成的方程式。 每个二次曲线都有重要的点和测量标准。 这些关键数字之间的关系被用来理解二次曲线是如何形成的,创建方程式,并绘制方程式的图表。

Chapter Summary
::章次摘要

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• Completing the square is a procedure that enables you to combine squared and linear terms of the same variable into a perfect square of a binomial.
  ::完成正方形是一个程序, 使您能够将同一个变量的平方和线性词合并为二进制的完美正方形 。
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• Conics are a family of graphs (not usually functions) that come from the same general equation. This family is the intersection of a double  cone and a plane in three-dimensional space.
  ::二次曲线是由来自相同一般方程式的图形( 通常不是函数) 组成的组合。 这个组合是双锥形和三维空间的平面的交叉点 。
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• The standard form of a conic is  $\begin{array}{r}A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0.\end{array}$
  ::二次曲线的标准形式为 Ax2+Bxy+Cy2+Dx+Ey+F=0。

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• A parabola is the collection of points that are equidistant from a fixed focus and directrix.
  ::抛物线是收集固定焦点和准点距离相等的点数。
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• The focus of a parabola is the point that the parabola seems to curve around.
  ::抛物线的焦点是,抛物线似乎在周围旋转。
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• The directrix of a parabola is the line that the parabola seems to curve away from.
  ::抛物线的准点是 抛物线似乎向外倾斜的线条。
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• The general equation for a parabola opening vertically is $\begin{array}{r}(x-h{)}^2=\pm 4p(y-k).\end{array}$
  ::垂直开口的抛物线的一般方程为(x-h)2+4p(y-k)。
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• The general equation for a parabola opening horizontally is $\begin{array}{r}(y-k{)}^2=\pm 4p(x-h).\end{array}$
  ::水平打开的抛物线的一般方程是 (y-k) 24p(x-h) 。

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• A circle is the collection of points that are equidistant from a given point.
  ::a 圆是指从给定点收集相等于某一点的点数。
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• The radius of a circle is the distance from the center of the circle to the outside edge.
  ::圆的半径是从圆的中心到外部边缘的距离。
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• The center of a circle is the point that defines the location of the circle.
  ::圆的中心是确定圆位置的点。
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• The general equation of a circle is $\begin{array}{r}(x-h{)}^2+(y-k{)}^2=r^2,\end{array}$  where $\begin{array}{r}(h,k)\end{array}$  is the center of the circle and r is the radius.
  ::圆的普通方程式是 (x-h) 2+(y-k)2=r2, 其中 k) 是圆的中心, r 是圆的半径。

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• 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。
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• 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。
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• 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:  $\begin{array}{r}e=\frac{c}{a}.\end{array}$
  ::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。
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• The major axis of an ellipse is its longer  line segment that runs through the center and both foci, with ends at the widest endpoints.
  ::椭圆的主要轴是其长线段,穿过中间和角,以最宽的终点为终点。
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• The semi-major axis is one half of the major axis, and thus runs from the center, through a focus, and to the perimeter. It is the longest radius of an ellipse.
  ::半主轴是主轴的一半,因此从中枢、一个焦点和周界运行。它是椭圆最长半径。
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• The minor axis of an ellipse is its shorter line segment, which runs through the center, with ends at the more narrow endpoints.
  ::椭圆的小轴是其短线段,穿过中心,以更窄的端点为终点。
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• The semi-minor axis is the shortest radius of an ellipse.
  ::半最小轴是椭圆的半径。

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• A hyperbola is the collection of points that share a constant difference between the distances between two focus points.
  ::双倍波拉是收集在两个焦点点之间的距离之间始终存在差异的点数。
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• Eccentricity is the ratio between the length of the focal radius and the length of the semi-transverse axis. For hyperbolas, the eccentricity is greater than 1.
  ::偏心度是焦半径长度与半反向轴长度之比。对于超光子,偏心度大于1。
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• The graphing form of a hyperbola that opens side to side is  $\begin{array}{r}\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1.\end{array}$
  ::向侧打开的双倍波拉的图形形式是 (x-h) 2a2 - (y-k) 2b2=1 。
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• The graphing form of a hyperbola that opens up and down is  $\begin{array}{r}\frac{(y-k{)}^2}{a^2}-\frac{(x-h{)}^2}{b^2}=1.\end{array}$
  ::向上和向下打开的双倍波拉的图形形式是 (y-k) 2a2 - (x-h) 2b2=1。

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• A degenerate conic is generated when a plane intersects the vertex of the cone .
  ::当平面交叉锥体的顶部时,就会产生退化的二次曲线。
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• There are three types of degenerate conics: a single point, a line or two parallel lines, or two intersecting lines.
  ::有三种退化的二次曲线:一点、一线或两条平行线,或两条交叉线。

Review
::回顾

Try the following cumulative review problems to practice the concepts we studied in this chapter:
::尝试下列累积审查问题,以实践我们在本章中研究的概念: