题目

Problem

The diagram shows a circle $C$ of radius $r$ , where $x > 0$ and $y > 0$ for all points on $C$ . The least distance between any point on $C$ and the $x$ -axis is 8 units, and the least distance between any point on $C$ and the $y$ -axis is 5 units.

(a) State the coordinates of the centre of the circle in terms of $r$ .

[1]

(b) Given that the distance between the origin and the centre of the circle is 15 units, find the value of $r$ .

[3]

Find the gradient of the tangent to the circle at $P$ .

[2]

题目中文翻译

图中显示了一个半径为 $r$ 的圆 $C$ ，并且圆 $C$ 上所有点都满足 $x > 0$ 、 $y > 0$ 。圆 $C$ 上任意一点到 $x$ 轴的最短距离为 8 个单位，到 $y$ 轴的最短距离为 5 个单位。

(a) 用 $r$ 表示圆心坐标。

[1]

(b) 已知原点到圆心的距离为 15 个单位，求 $r$ 的值。

[3]

求圆在 $P$ 点处切线的斜率。

[2]

解答

(a)

解法一

思路

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圆心到任一坐标轴的距离，比圆上最近点到该轴的距离多一个半径。横坐标对应到 $y$ 轴的距离。

答题过程

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The least $x$ -coordinate is $5$ and the least $y$ -coordinate is $8$ . Adding $r$ to each gives the centre $(r+5,r+8)$ .

(b)

解法一

思路

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把 (a) 的圆心坐标代入两点距离公式。半径必须为正。

答题过程

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The centre distance is $15$ , so

(r+5)^2+(r+8)^2=225 \quad\Longrightarrow\quad r^2+13r-68=(r+17)(r-4)=0.

Thus $r=4$ , since the other root is negative.

(c)

解法一

思路

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离原点最远的点在原点到圆心的射线上，过该点的半径与这条射线同方向。

答题过程

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The farthest point lies on the ray from the origin through the centre, where the distance from the origin reaches $15+r$ . The centre is $(9,12)$ , so this radius has slope $12/9=4/3$ . Its perpendicular tangent has gradient $-\frac34$ .

CIE 9709 2025 March Paper 12 Q6

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

(c)

解法一

思路

答题过程