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  • Hyperbolas 和 Asymptotes 和亚丁

    Like other , can be created by "slicing" a cone and looking at the cross-section. Unlike other conics, hyperbolas actually require 2 cones stacked on top of each other, point to point. The shape is the result of effectively creating a parabola out of both cones at the same time.
    ::像其他的一样,可以通过“切除”锥形和截面来创造。与其他二次二次曲线不同,超光子实际上需要两个相互叠叠的锥体,点对点。形状是同时从两个锥体中有效产生抛物线的结果。

    So the question is, should hyperbolas really be considered a shape all their own? Or are they just two graphed at the same time? Could "different" shapes be made from any of the other conic sections if two cones were used at the same time?
    ::那么问题是,超光子是否真的被看作一种形状? 或者它们只是同时用两个图表来表示?如果同时使用两个锥形,其他二次曲线的形状是否会“不同”呢?

    Hyperbolas and Asymptotes
    ::Hyperbolas 和 Asymptotes 和亚丁

    In addition to their focal property, hyperbolas also have another interesting geometric property. Unlike a parabola, a hyperbola becomes infinitesimally close to a certain line as the x − or y − coordinates approach infinity. What we mean by “infinitesimally close?” Here we mean two things: 1) The further you go along the curve, the closer you get to the asymptote, and 2) If you name a distance , no matter how small, eventually the curve will be that close to the asymptote. Or, using the language of limits, as we go further from the vertex of the hyperbola the limit of the distance between the hyperbola and the asymptote is 0.
    ::除了焦点属性外,超光谱还具有另一个有趣的几何属性。 与抛物线不同, 超光波体会变得极小地接近某条线, 因为它是x- 或 y- 坐标法的无限性。 我们的意思是“ 最接近? ” 这里我们的意思是两件事sad1) 沿着曲线走得越远, 你越接近无线点;(2) 如果你说出一个距离, 不论距离多小, 曲线最终会接近无线点。 或者, 当我们从超光波体的脊椎走得越远时, 超光波体和无线之间的距离限制是0 。

    These lines are called asymptotes .
    ::这些直线被称为“静态”线。

    There are two asymptotes, and they cross at the point at which the hyperbola is centered:
    ::有两个小粒子, 它们穿过超重波拉的中心点:

    For a hyperbola of the form x 2 a 2 − y 2 b 2 = 1 , the asymptotes are the lines:
    ::对于表x2a2-y2b2=1的双倍波拉,单位数为线条:

    y = b a x and y = − b a x .
    ::y'bax和y'bax。 y'bax和y'bax。

    For a hyperbola of the form y 2 a 2 − x 2 b 2 = 1 the asymptotes are the lines:
    ::y2a2- x2b2=1 的双倍波拉,其线条为:

    y = a b x and y = − a b x .
    ::y=abx和yabx。

    (For a shifted hyperbola, the asymptotes shift accordingly.)
    :sad对于变换的双倍波拉,小微粒也相应变换。 )

    Examples
    ::实例

    Example 1
    ::例1

    Earlier, you were asked if hyperbolas should be considered shapes all their own.
    ::早些时候,有人问过你 是否超黄球应该 被认为是形状 所有的他们自己的。

    Hyperbolas are considered different shapes, because there are specific behaviors that are unique to hyperbolas. Also, though hyperbolas are the result of dual parabolas, none of the other conics really create unique shapes with dual cones - just double figures - and in any case require multiple "slices".
    ::超氯代谢物被认为是不同的形状,因为有些特定的行为是超氯代谢物所特有的。 此外,虽然超氯代谢物是双倍副苯的产物,但其他的二次二次曲线都没有真正创造出具有双锥形的独特形状 — — 仅仅是双倍数字 — — 并且无论如何需要多重“切片 ” 。

    Example 2
    ::例2

    Graph the following hyperbola, drawing its foci and asymptotes and using them to create a better drawing: 9 x 2 − 36 x − 4 y 2 − 16 y − 16 = 0 .
    ::绘制以下双曲线图,绘制其角和微粒,并用它们绘制更好的绘图:9x2-36x-4y2-16y-16y-16=0。

    First, we put the hyperbola into the standard form:
    ::首先,我们把超波拉 变成标准形式:

    9 ( x 2 − 4 x ) − 4 ( y 2 + 4 y ) = 16 9 ( x 2 − 4 x + 4 ) − 4 ( y 2 + 4 y + 4 ) = 36 ( x − 2 ) 2 4 − ( y + 2 ) 2 9 = 1

    ::9(x2-4x)-4(y2+4y)=169(x2-4x+4)-4(y2+4y+4)=36(x-2)24-(y+2)29=1

    So a = 2 , b = 3 and c = 4 + 9 = 13 . The hyperbola is horizontally oriented, centered at the point (2,-2), with foci at ( 2 + 13 , − 2 ) and ( 2 − 13 , − 2 ) . After taking shifting into consideration, the asymptotes are the lines: y + 2 = 3 2 ( x − 2 ) and y + 2 = − 3 2 ( x − 2 ) . So graphing the vertices and a few points on either side, we see the hyperbola looks something like this:
    ::a=2, b=3 和 c=4+9=13。 双曲线以水平为方向, 以点为中心(2, 2), 角值为(2+13, -2) 和(2 - 13, -2) 。 在考虑到变化后, 偏移的线是: y+2=32(x-2) 和 y+2+32(x-2) 。 因此, 将两侧的顶端和几个点划成图表, 我们可以看到双曲线是像这样的 :

    Example 3
    ::例3

    Graph the following hyperbola, drawing its foci and asymptotes and using them to create a better drawing: 16 x 2 − 96 x − 9 y 2 − 36 y − 84 = 0 .
    ::绘制以下双曲线图,绘制其角和微粒,并用它们绘制更好的绘图:16x2-96x-9y2-36y-84=0。

    Example 4
    ::例4

    Graph the following hyperbola, drawing its foci and asymptotes, and use them to create a better drawing: y 2 − 14 y − 25 x 2 − 200 x − 376 = 0 .
    ::绘制以下的双曲线图,绘制其角和微粒,并用来绘制更好的图象:y2-14y-25x2-200x-376=0。

    Example 5
    ::例5

    Find the equation for a hyperbola with asymptotes of slopes 5 12 and − 5 12 , and foci at points (2,11) and (2,1).
    ::查找具有512和-512个斜坡和512个斜坡及2、11和2、1点方子等离子的超重波拉方程。

    ( y − 6 ) 2 25 − ( x − 2 ) 2 144 = 1
    :sady-6)225-(x-2-2)214=1

    Example 6
    ::例6

    A hyperbola with perpendicular asymptotes is called perpendicular . What does the equation of a perpendicular hyperbola look like?
    ::带有垂直心血管的双波拉叫做垂直心血管。垂直心血管超波拉的方程式长什么样?

    The slopes of perpendicular lines are negative reciprocals of each other. This means that a b = b a , which, for positive a and b means a = b .
    ::垂直线的斜坡是相互负对等的。 这意味着bab=ba, 这表示阳性。

    Example 7
    ::例7

    Find an equation of the hyperbola with x-intercepts at x = –7 and x = 5, and foci at (–6, 0) and (4, 0).
    ::在x = -7 和 x = 5 和 foci 值 ( - 6, 0) 和 (4, 0) 和 (4, 0) 上找到带有 x 界面的双倍波拉方程。

    The foci have the same y-coordinates, so this is a left/right hyperbola with the center , foci, and vertices on a line paralleling the x-axis.
    ::方形具有相同的 Y 坐标, 所以这是一个左/ 右双曲线, 中间、 方形和脊椎在与 X 轴平行的线上 。

    Since it is a left/right hyperbola, the y part of the equation will be negative and equation will lead with the x 2 term (since the leading term is positive by convention and the squared term must have different signs if this is a hyperbola).: The center is midway between the foci, so the center ( h , k ) = ( − 1 , 0 ) . The foci c are 5 units to either side of the center, so c = 5 → c 2 = 25 .
    ::由于它是左/右双曲线, 正方程的y 部分将是负的, 而正方程将随着x2 术语( 因为根据公约, 领先术语是正的, 平方术语必须具有不同的信号 ) 。 : 中心位于方块中间, 所以中心 (h, k) = (-1, 0) 。 方程的 foci c 是中间两侧的 5 个单位, 所以 c= 5c2= 25 。

    The x-intercepts are 4 units to either side of the center, and the foci are on the x-axis so the intercepts must be the vertices a a = 4 → a 2 = 16 .
    ::中间两侧的 X 截取器是 4 个单位, 方轴在 X 轴上, 所以拦截必须是 an= 4 a2= 16 的顶部 。

    Use the Pythagorean theorem, a 2 + b 2 = c 2 , to get b 2 = 25 − 16 = 9 .
    ::使用毕达哥里定理, a2+b2=c2, 以获得 b2=25-16=9。

    Substitute the calculated values into the standard form ( x − h ) 2 a − ( y − k ) 2 b = 1 to get  ( x + 1 ) 2 16 − y 2 9 = 1 .
    ::以标准表格(x-h)2a-(y-k)2b=1 的计算值替代标准表格(x+1)216-y29=1。

    Review
    ::回顾

    Find the equations of the asymptotes of each hyperbola.
    ::查找每个超重波的微粒方程式的方程。

    1. ( y + 3 ) 2 4 − ( x − 2 ) 2 = 1
      :sady+3)24-(x-2)2=1
    2. y 2 16 − ( x + 3 ) 2 = 1
      ::y216 - (x+3) 2=1
    3. ( x + 2 ) 2 4 − ( y + 1 ) 2 9 = 1
      :sadx+2)24-(y+1)29=1
    4. ( y − 4 ) 2 16 − ( x − 4 ) 2 16 = 1
      :sady-4)216-(x-4)216=1
    5. ( x − 1 ) 2 1 − 9 ( y + 4 ) 2 1 = 9
      :sadx-1)21-9(y+4)21=9
    6. ( y + 2 ) 2 16 − ( x − 2 ) 2 1 = 1
      :sady+2)216-(x-2)21=1
    7. ( x − 4 ) 2 1 − ( y + 1 ) 2 4 = 1
      :sadx-4)21-(y+1)24=1
    8. y 2 16 − ( x + 1 ) 2 4 = 1
      ::y216 - (x+1) 24=1
    9. ( x − 3 ) 2 4 − ( y − 4 ) 2 1 = 1
      :sadx-3)24-(y-4)21=1
    10. ( x − 4 ) 2 4 − ( y − 3 ) 2 1 = 1
      :sadx-4)24-(y-3)21=1

    Graph the hyperbolas, give the equation of the asymptotes, and use the asymptotes to enhance the accuracy of your graph.
    ::绘制双倍数图,给出小数点的方程式,并使用小数点来提高图形的准确性。

    1. ( x + 4 ) 2 4 − ( y − 1 ) 2 9 = 1
      :sadx+4)24-(y-1)29=1
    2. ( y + 3 ) 2 4 − ( x − 4 ) 2 9 = 1
      :sady+3)24-(x-4)29=1
    3. ( y + 4 ) 2 16 − ( x − 1 ) 2 4 = 1
      :sady+4) 216-(x-1)24=1
    4. ( x − 2 ) 2 − 4 y 2 = 16
      :sadx-2)2-2-4y2=16
    5. y 2 4 − ( x − 1 ) 2 4 = 1
      ::y24 - (x- 1) 24= 1
    6. ( x − 2 ) 2 16 − ( y + 4 ) 2 1 = 1
      :sadx-2)216-(y+4)21=1
    7. ( x + 2 ) 2 9 − ( y + 2 ) 2 16 = 1
      :sadx+2)29-(y+2)216=1
    8. ( x + 4 ) 2 9 − ( y − 2 ) 2 4 = 1
      :sadx+4)29-(y-2)24=1

    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 。