이차곡선, 원의 관계 및 기하학적 측량
ACT 주제 15 5:10 영어 내레이션 · 영어 + 중국어 자막 burned-in
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전사본
An ellipse, x minus h squared over a squared plus y minus k squared over b squared equals 1, has centre at h comma k.
椭圆 (x−h)²/a² + (y−k)²/b² = 1 的中心为 (h, k)。
The semiaxes are the square roots of the positive denominators.
半轴长为正分母的平方根。
A parabola, y squared equals 4 p x, has focus at p comma 0 and directrix x equals negative p.
抛物线 y² = 4px 的焦点为 (p, 0)，准线为 x = −p。
Distinguish these equations from a hyperbola with a difference of squares.
注意将这些方程与具有平方差形式的双曲线方程区分开来。
Each task gives its own data.
每个任务都有各自的数据。
Do not copy the worked example’s numbers.
请勿照搬例题中的数值。
Pause here and attempt the task before revealing the reasoning.
请先暂停，完成这道原创练习再继续。
Use the full context on the preceding pages.
请利用前几页的完整背景信息。
The centre is negative 2 comma 1, the semiaxes are 5 and 3, and the endpoints are negative 7 comma 1, 3 comma 1, negative 2 comma negative 2, and negative 2 comma 4.
中心是（负二，一），半轴长是五和三，四个端点是（负七，一）、（三，一）、（负二，负二）和（负二，四）。
In the parabola, 4 p equals negative 12, so p equals negative 3; the vertex is 0 comma 0, and the focus is negative 3 comma 0.
在抛物线中，四 p 等于负十二，所以 p 等于负三；顶点是（零，零），焦点是（负三，零）。
Pause here and attempt the task before revealing the reasoning.
请先暂停，完成这道原创练习再继续。
Use the full context on the preceding pages.
请结合前几页的完整背景信息。
Use the tangent-secant relation, t squared equals e times w.
使用切线与割线的关系：t 的平方等于 e 乘 w。
Rearranged, w equals t squared over e, which is 6 squared over 4, or 9 centimetres.
变形得 w 等于 t 平方除以 e，即六的平方除以四，等于九厘米。
The inside length is w minus e, 9 minus 4, which is 5 centimetres.
圆内部分的长度是 w 减 e，九减四，等于五厘米。
The product uses the whole length, not just the inside part.
乘积用的是整条割线的长度，而不只是圆内部分。
Pause here and attempt the task before revealing the reasoning.
请先暂停，完成这道原创练习再继续。
Use the full context on the preceding pages.
使用前面页面的完整背景信息。
The volume is pi r squared h, which is pi times 3 squared times 8, or 72 pi cubic centimetres.
体积等于 π r 平方 h，即 π 乘三的平方乘八，等于七十二 π 立方厘米。
The surface area is 2 pi r squared plus 2 pi r h, which is 2 pi times 3 squared, plus 2 pi times 3 times 8, giving 66 pi square centimetres.
表面积等于二 π r 平方加二 π r h，即二 π 乘三的平方加二 π 乘三乘八，等于六十六 π 平方厘米。
Doubling all lengths multiplies the volume by 8 and the area by 4.
所有长度都加倍时，体积乘以八，面积乘以四。
Pause here and attempt the task before revealing the reasoning.
请先暂停，完成这道原创练习再继续。
Use the full context on the preceding pages.
请利用前几页的完整背景信息。
The difference of squared terms defines a hyperbola centred at the origin.
平方项相减定义的是以原点为中心的双曲线。
The sum with equal coefficients defines a circle of radius 4.
系数相等的平方项相加定义的是半径为四的圆。
Its circumference is 2 pi r, which is 8 pi, and its area is pi r squared, which is 16 pi, in coordinate length and square-length units respectively.
它的周长是二 π r，即八 π，面积是 π r 平方，即十六 π，单位分别是坐标长度和坐标面积。
Secant products match, e 1 w 1 equals e 2 w 2.
两条割线的乘积相等：e1 乘 w1 等于 e2 乘 w2。
So w 2 equals 3 times 12, over 4, which is 9 centimetres; the inside length is 9 minus 4, which is 5 centimetres.
所以 w2 等于三乘十二除以四，即九厘米；圆内部分是九减四，等于五厘米。
Outside times inside would apply the wrong relation.
用“圆外部分乘圆内部分”就用错了关系。
Return to the task and explain why the method fits.
回到该任务，并解释为何此方法适用。
Original practice does not predict an official score.
原始练习无法预测正式成绩。
Complete required live or recorded evidence separately.
需单独完成所需的现场或录制证据。
Pause and test yourself. Cover the worked answer, rebuild the answer to the last example in your own words, and name one condition that would change it before you check.
暂停一下，自我检测：遮住示范答案，用自己的话重做最后一个例子的答案，并在核对之前说出一个会改变结论的条件。