题目
The diagram shows a curve which has a maximum point at and a minimum point at . The curve is the result of applying a combination of two transformations to a circle. The first transformation applied is a translation of . The second transformation applied is a stretch in the -direction.
(a) State the scale factor of the stretch.
(b) State the radius of the original circle.
(c) State the coordinates of the centre of the circle after the translation has been completed but before the stretch is applied.
(d) State the coordinates of the centre of the original circle.
题目中文翻译
图中曲线的最高点为 ，最低点为 。这条曲线是将一个圆依次进行两个变换得到的。第一个变换是按向量 平移；第二个变换是沿 方向拉伸。
(a) 写出拉伸的比例因子。
(b) 写出原来圆的半径。
(c) 写出完成平移后、进行拉伸前的圆心坐标。
(d) 写出原来圆的圆心坐标。
原卷通用作答要求
Answer all questions. You should use a calculator where appropriate.
You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。
解答
(a)
解法一
思路
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纵向拉伸只改变竖直直径。把变换后的竖直直径与仍保持原长的水平直径相比。
答题过程
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The vertical diameter is . The horizontal diameter, unchanged by the stretch, is . The scale factor is .
(b)
解法一
思路
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水平宽度在两次变换下都不变，因此它的一半就是原圆半径。
答题过程
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The original radius is .
(c)
解法一
思路
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由最高点和最低点找椭圆中心，再撤销纵向拉伸。
答题过程
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The transformed centre is . Dividing its height by gives the centre before the stretch.
(d)
解法一
思路
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从 (c) 的圆心减去平移向量。
答题过程
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Reversing the translation gives .