Section: 结结锥部分的一般形式 | 6.9 锥形锥形块的一般形式

Section outline

Kelli was reviewing for her pre-calc final on , and she was trying to memorize all of the rules for each of the shapes. She was actually doing pretty well, but was frustrated with the challenge of keeping them all straight.
::Kelli正在审查她的计算前决赛, 她试图记住每个形状的所有规则。她实际上做的很好,但对于保持它们保持直线的挑战感到沮丧。

Kelli found herself wishing there were a way to identify the shape early on, so she would have an idea of what she was generally sketching before she worried about specifics.
::Kelli发现自己希望有办法早期确定形状所以在她担心细节之前她会知道她一般在画什么

Too bad Kelli didn't know about the material in this lesson!
::可惜Kelli不知道这堂课的内容!

General Forms of Conic Sections
::结结锥部分的一般形式

Let’s examine all the equations of the conic sections we’ve studied in this chapter.
::让我们审视我们在本章中所研究的二次曲线部分的所有方程式。

: $\begin{aligned} \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ are any positive numbers (the circle is the specific case when $\begin{aligned} a = b \end{aligned}$ ).
::: x2a2+y2b2=1,其中a和b为任何正数(a=b为圆)。

: $\begin{aligned} y = a x^{2} \end{aligned}$ or $\begin{aligned} x = a y^{2} \end{aligned}$ where $\begin{aligned} a \end{aligned}$ is any non-zero number.
:::y=ax2 或 x=ay2,其中一个数字为非零数字。

: $\begin{aligned} \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \end{aligned}$ or $\begin{aligned} \frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1 \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ are any positive numbers.
::: x2a2-y2b2=1 或 y2a2-x2b2=1,其中a和b为正数。

All these equations have in common that they are degree-2 polynomials, meaning the highest exponent of any variable—or sum of exponents of products of variables—is two. So for example, here are some degree two polynomial equations in a more general form:
::所有这些方程式的共同点是,它们都是程度-2多义方程式,意思是任何变量——或变量产品出处的总和——最高的指数是两个变量。例如,这里有某种程度的2个多义方程式,其形式更为笼统:

$\begin{aligned} 2 x^{2} + 5 y^{2} - 3 y + 4 & = 0 \\ x^{2} - 3 y^{2} + x - y + 3 & = 0 \\ 10 x^{2} - y - 5 & = 0 \\ x y & = 2 \end{aligned}$

::2x2+5y2-3y+4=0x2-3y2+x-y3=010x2-y-5=0xy=2

Some of these probably already look like conic sections to you. For example, in the first equation, we can complete the square to remove the $\begin{aligned} - 3 y \end{aligned}$ term and we will see that we have an ellipse. In the second equation we can complete the square twice to remove both the $\begin{aligned} x \end{aligned}$ and $\begin{aligned} - y \end{aligned}$ terms and we will have a hyperbola. This is a hyperbola, not an ellipse, because the coefficient of the $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ terms have opposite signs.
::其中一些可能已经看起来像二次曲线。例如,在第一个方程中,我们可以完成方形去掉- 3y 术语,我们会看到我们有一个椭圆。在第二个方形中,我们可以完成两次方形去掉 x 和 y 术语,我们会有一个双倍的双倍值。这是一个双倍值,而不是椭圆值,因为 x2 和 y2 术语的系数有相反的符号。

The third equation is a parabola since there is an $\begin{aligned} x^{2} \end{aligned}$ term and $\begin{aligned} y \end{aligned}$ term but not a $\begin{aligned} y^{2} \end{aligned}$ term. Do you see how you can solve for $\begin{aligned} y \end{aligned}$ , putting the equation in the standard form for a vertically oriented parabola?
::第三个方程是一个抛物线, 因为有一个 x2 术语和y 术语, 但不是一个 y2 术语。您看到如何解决 y , 将公式放在垂直方向的抛物线的标准格式中 ?

But what about the fourth equation? Like the others, it is a degree-2 polynomial, since the exponents of the $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ term sum to 2. But the fourth equation looks nothing like any of the forms for conic sections that we’ve examined so far. Nonetheless, as we saw in the last section, $\begin{aligned} x y = 2 \end{aligned}$ appears to be a hyperbola with foci (2,2) and (-2,-2). The reason it doesn’t fit either of the standard forms for hyperbolas is because it is diagonally oriented, rather than horizontally or vertically oriented (do you see how its two foci lie on a diagonal line, rather than a horizontal or vertical line?)
::但第四等式又如何呢?与其他方程式一样,它是一个度-2多面方程式,因为x和y术语的表率是 2 。但第四等式看起来与我们迄今所检查的二次线段的表状完全不一样。然而,正如我们在最后一节所看到的那样,xy=2似乎是一个带有 foci (2,2) 和 (2,2) 的双面体。它与双面体标准格式中任何一个格式不匹配的原因是它具有对角方向,而不是水平或垂直方向(你看到它的两个顶部位于对角线上,而不是水平或垂直线吗? )

In order to see how such differently-oriented conic sections fit into our standard forms, we need to rotate them so that they are either horizontally or vertically oriented.
::为了了解这种不同方向的二次曲线部分如何适合我们的标准格式,我们需要将其旋转,以便横向或纵向地取向。

Identifying Conics
::确定二次曲线

How can we look at a degree-2 polynomial equation and determine which conic section it depicts?
::我们如何看待度2 -2多面方程, 确定它所描述的二次曲线部分?

$\begin{aligned} A x^{2} + B y^{2} + C x y + D x + E y + F = 0 \end{aligned}$

::Ax2+By2+Cxy+Dx+Ey+F=0 Ax2+By2+Cxy+Dx+Ey+F=0

When $\begin{aligned} C = 0 \end{aligned}$ we have already discussed how to determine which conic section the equation refers to. In summary, if $\begin{aligned} A \end{aligned}$ and $\begin{aligned} B \end{aligned}$ are both positive, the conic section is an ellipse. This is also true of $\begin{aligned} A \end{aligned}$ and $\begin{aligned} B \end{aligned}$ are both negative, as the entire equation can be multiplied by -1 without changing the solution set. If $\begin{aligned} A \end{aligned}$ and $\begin{aligned} B \end{aligned}$ differ in sign, the equation is a hyperbola, and if $\begin{aligned} A \end{aligned}$ or $\begin{aligned} B \end{aligned}$ equals zero the equation is a parabola.
::当 C=0 我们已经讨论过如何确定方程式指的是哪个二次曲线部分时。总之, 如果 A 和 B 都为正数, 二次曲线部分是一个椭圆。 A 和 B 也是负数, 因为整个方程式可以乘以 - 1 而不改变设定的解决方案。如果 A 和 B 的符号不同, 方程式是双曲线, 如果 A 或 B 等于零, 方程式是 parbola 。

There are a few new, more general, rules I will show you that give more information about the case when $\begin{aligned} C \neq 0 \end{aligned}$ and hence the conic section needs to be rotated to achieve horizontal or vertical orientation.
::我将向你们展示一些新的、更一般性的规则,其中将提供更多关于C0和二次曲线部分需要旋转以达到横向或纵向方向的情况的信息。

If $\begin{aligned} C^{2} < 4 A B \end{aligned}$ , the equation is an ellipse (note when $\begin{aligned} C = 0 \end{aligned}$ this holds whenever $\begin{aligned} A \end{aligned}$ and $\begin{aligned} B \end{aligned}$ are the same sign, which is consistent with our simpler rule stated above.)
::如果C2 < 4AB, 方程为椭圆(注意当C=0时, 只要A和B是同一种符号, 即保持此值, 这符合上述更简单的规则。 )

If $\begin{aligned} C^{2} > 4 A B \end{aligned}$ , the equation is an hyperbola (note when $\begin{aligned} C = 0 \end{aligned}$ this holds whenever $\begin{aligned} A \end{aligned}$ and $\begin{aligned} B \end{aligned}$ are the opposite sign, which is consistent with our simpler rule stated above.)
::如果C2>4AB,则方程式为超重波拉(注意当C=0时,当A和B是相反的符号时,它会保持,这符合上述更简单的规则。 )

If $\begin{aligned} C^{2} = 4 A B \end{aligned}$ , the equation is a parabola (note when $\begin{aligned} C = 0 \end{aligned}$ , either $\begin{aligned} A \end{aligned}$ or $\begin{aligned} B \end{aligned}$ equals zero, which is consistent with our simpler rule stated above.)
::如果C2=4AB,则方程为抛物线(注意,当C=0, A或B等于零时,这符合上述更简单的规则)。 )

Rotating Conics
::旋转二次曲线

Suppose we have an equation of degree two polynomials, such as the $\begin{aligned} x y = 2 \end{aligned}$ example discussed above. In order to put it into the more recognizable form of a ellipse, parabola, or a hyperbola, we need to rotate it in such a way so that rotated version has no $\begin{aligned} x y \end{aligned}$ term. So we need to find an appropriate angle $\begin{aligned} θ \end{aligned}$ such that changing the $\begin{aligned} x - \end{aligned}$ coordinate to $\begin{aligned} x^{'} \cos (θ) - y^{'} \sin (θ) \end{aligned}$ and the $\begin{aligned} y - \end{aligned}$ coordinate to $\begin{aligned} x^{'} \sin (θ) + y^{'} \cos (θ) \end{aligned}$ results in an equation with no $\begin{aligned} x y \end{aligned}$ term.
::假设我们有一个度为2的多元等式, 如上文讨论的 Xy=2 示例。为了将其置于更可识别的椭圆形、 parbola 或 hybola 形式中, 我们需要以这样的方式旋转它, 这样旋转的版本没有xy 术语。所以我们需要找到一个合适的角度。因此我们需要找到一个合适的角度 , 来改变 x- 坐标到 x cos {\\ - yäsin { 和 y- 坐标到 x sinn { + y\ cos\ { 。

Examples
::实例

Example 1
::例1

State what type of conic section is represented by the following equation: $\begin{aligned} 5 x^{2} + 6 y^{2} + 2 x - 5 y + x y = 0 \end{aligned}$ .
::说明以下列方程式表示的二次曲线部分类型: 5x2+6y2+2x-5y+xy=0。

This is an ellipse, as both a and b are positive. Also, note that c < 4ab .
::这是一个椭圆形,因为a和b都是正数。另请注意 c < 4ab。

Example 2
::例2

State what type of conic section is represented by the following equation: $\begin{aligned} x^{2} + 3 y - 20 x y + 20 = 0 \end{aligned}$ .
::说明以下列方程式表示的二次曲线部分类型: x2+3y-20xy+20=0。

This is a hyperbola, since c ² > 4ab (400 > 12).
::这是一个双倍波拉,因为C2 > 4ab(400 > 12)。

Example 3
::例3

Which conic section is described by the following equation: $\begin{aligned} 4 x^{3} - 2 x^{2} + 37 = 0 \end{aligned}$ ?
::以下方程式描述了哪个二次曲线区域: 4x3-2x2+37=0?

To identify $\begin{aligned} 4 x^{3} - 2 x^{2} + 37 = 0 \end{aligned}$ we just need to look at the signs of the squared terms:
::为了确定 4x3 - 2x2+37=0,我们只需要看看平方字的符号:

The $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ coefficients have different signs so it's a hyperbola.
::x2和y2系数有不同的符号所以它是一个双倍波拉。

Additionally, the $\begin{aligned} y^{2} \end{aligned}$ term is negative, so it has a vertical transverse.
::此外,Y2一词是阴性的,因此具有垂直反向。

For the following examples, use the information below:
::对于以下例子,请使用以下信息:

When identifying conic sections, the squared terms are the only factors that matter. Run the standard form of the equation you are questioning through the following series of tests on the squared terms:
::当识别二次曲线区域时, 方形术语是唯一重要的因素。运行您询问的方程式的标准形式, 通过以下一系列对二次曲线条件的测试 :

If only 1 variable is squared, you have a parabola, stop further testing.
::如果只有1个变量方形,您有抛物线,请停止进一步测试。

If the coefficients of the squared terms have opposite signs, you have a hyperbola, stop further testing.
::如果平方条件的系数有相反的符号, 您有一个超重波拉, 停止进一步测试。

If the squared terms are multiplied by the same coefficient, you have a circle, stop testing.
::如果平方条件乘以相同系数,则您有一个圆圈,停止测试。

If none of the above apply, you have an ellipse.
::如果没有上述任何一项适用,您就有一个椭圆。

Example 4
::例4

Identify and describe the conic section represented by the following equation: $\begin{aligned} (x - 1)^{2} + (y + 1)^{2} = 9 \end{aligned}$ .
::识别和描述以下列方程式表示的二次曲线部分x-1)2+(y+1)2=9。

Both terms are squared: not a parabola
::两个词都平方:不是抛物线

The coefficients of $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ are both positive: not a hyperbola
::x2 和 y2 的系数都是正数: 不是双倍值

Both squared term coefficients are the same: a circle
::两个平方词系数相同:圆

$\begin{aligned} r^{2} = 9 \to r = 3 \end{aligned}$
::r2=9°r=3

The center $\begin{aligned} (h, k) \end{aligned}$ is $\begin{aligned} (1, - 1) \end{aligned}$ .
::中心(h,k)为(1,-1)。

Example 5
::例5

State what type of conic section is identified by the following equation: $\begin{aligned} - 8 x^{2} - 18 y^{2} + 48 x - 360 y - 1800 = 0 \end{aligned}$ .
::说明以下列方程式确定二次曲线部分的类型:-8x2-18y2+48x-360y-1800=0。

$\begin{aligned} - 8 (x - 3)^{2} - 18 (y + 10)^{2} = - 72 \end{aligned}$ ..... Complete the square
::-8(x-3-3)2-18(y+10)2 72...。完成正方形

$\begin{aligned} \frac{(x - 3)^{2}}{9} + \frac{(y + 10)^{2}}{4} = 1 \end{aligned}$ ..... Simplify
:x-3)29+(y+10)24=1 ...。简化

Both $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ are squared: not a parabola
::x 和 y 均平方:不是抛物线

Both coefficients are negative: not a hyperbola
::这两个系数都是负的:不是超重系数。

The $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ terms do not have the same coefficients: not a circle either
::x2 和 y2 条件的系数不相同:也不是圆

This ellipse has center: (3, -10)
::这个椭圆有中心( 3, - 10) 。

From the center, plot 2 up and $\begin{aligned} \sqrt{9} \end{aligned}$ left and right to mark the vertices.
::从中间,图2向上和9向左和右标记脊椎。

Example 6
::例6

State and describe the conic section: $\begin{aligned} 2 x^{2} - 16 x + 32 + 3 y = 6 \end{aligned}$ .
::状态并描述二次曲线部分: 2x2-16x+32+3y=6。

Only the $\begin{aligned} x \end{aligned}$ term is squared, this is a parabola
::只有 x 术语是方形, 这是抛物线

$\begin{aligned} 2 (x - 4)^{2} + 3 (y - 2) = 0 \end{aligned}$ ..... Factor and simplify
::2(x-4)2+3(y-2)=0.。

The vertex $\begin{aligned} (h, k) \end{aligned}$ is $\begin{aligned} (4, 2) \end{aligned}$ .
::顶点(h,k)为(4,2) 。

The $\begin{aligned} x \end{aligned}$ term is squared, this is an up/down parabola
::x 术语是平方, 这是向上/ 向下抛物线

The $\begin{aligned} x^{2} \end{aligned}$ term is positive, this parabola opens down
::x2 术语为正, 此抛物体向下打开

The parent $\begin{aligned} y = x^{2} \end{aligned}$ has been compressed vertically by 3 and horizontally by 2, resulting in a slightly wider than standard curve.
::母 y=x2 垂直压缩为3, 水平压缩为2, 其结果略大于标准曲线。

Review
::回顾

Identify the conic that is represented by the equation.
::确定方程式所代表的二次曲线。

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

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

$\begin{aligned} x^{2} - 5 x - y^{2} - 4 y + 16 = 0 \end{aligned}$
::x2 - 5x- y2 - 4y+16=0

$\begin{aligned} \frac{2 y}{3} = 2 (x - 4)^{2} + 1 \end{aligned}$
::2y3=2(x-4)2+1

Convert to standard form:
::转换为标准表单 :

$\begin{aligned} - 8 x^{2} - 18 y^{2} + 48 x - 360 y - 1800 = 0 \end{aligned}$
::-8x2-18y2+48x-360y-1800=0

$\begin{aligned} - 12 x^{2} + 12 y^{2} - 72 x + 192 y + 708 = 0 \end{aligned}$
::- 12x2+12y2 - 72x+192y+708=0

$\begin{aligned} x^{2} + y^{2} - 8 x + 10 y + 32 = 0 \end{aligned}$
::x2+y2-8x+10y+32=0

$\begin{aligned} 9 x^{2} + 4 y^{2} - 36 x + 64 y + 256 = 0 \end{aligned}$
::9x2+4y2 - 36x+64y+256=0

$\begin{aligned} 9 x^{2} + y^{2} + 90 x - 8 y + 232 = 0 \end{aligned}$
::9x2+y2+90x-8y+232=0

$\begin{aligned} 9 x^{2} - 9 y^{2} + 162 x + 162 y - 81 = 0 \end{aligned}$
::9x2--9y2+162x+162y-81=0

Identify and graph the following:
::确定并绘制如下图表:

%5E2%20%3D%201"> $\begin{aligned} \frac{(x + 4)^{2}}{9} + (y)^{2} = 1 \end{aligned}$
:x+4)29+2=1

$\begin{aligned} y = - 2 (x + 4)^{2} - 4 \end{aligned}$
::y2 (x+4) 2 - 4

$\begin{aligned} (x + 2)^{2} - (y - 2)^{2} = 1 \end{aligned}$
:x+2)2-(y-2)2=1

%5E2%7D%7B16%7D%20%3D%201%20"> $\begin{aligned} \frac{(x - 1)^{2}}{4} + \frac{(y)^{2}}{16} = 1 \end{aligned}$
:x-1)24+216=1

$\begin{aligned} x = 2 (y + 2)^{2} - 1 \end{aligned}$
::x=2(y+2)2-2-1

$\begin{aligned} x = - (y - 2)^{2} + 4 \end{aligned}$
::x(y-2)2+4

$\begin{aligned} \frac{(x + 2)^{2}}{1} - \frac{(y - 2)^{2}}{1} = 1 \end{aligned}$
:x+2)21-21-(y-2)21=1

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

Section outline

General Forms of Conic Sections ::结结锥部分的一般形式

Identifying Conics ::确定二次曲线

Rotating Conics ::旋转二次曲线

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)

General Forms of Conic Sections
::结结锥部分的一般形式

Identifying Conics
::确定二次曲线

Rotating Conics
::旋转二次曲线

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Review
::回顾

Review (Answers)
::回顾(答复)