题目

Problem

The line joining the points $(-2,5)$ and $(4,15)$ is the diameter of a circle $C$ .

(a) Find an equation for $C$ .

(5)

(b) Hence find the exact coordinates of the point on $C$ that is nearest the $x$ -axis.

(2)

解答

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直径两端点已知。圆心是直径中点，半径平方可以用圆心到任一端点的距离平方来求。

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The centre is the midpoint of $(-2,5)$ and $(4,15)$ :

\begin{align*} \left(\frac{-2+4}{2},\frac{5+15}{2}\right) =&\,(1,10). \end{align*}

The radius squared is

\begin{align*} r^2=&\,(4-1)^2+(15-10)^2\\ =&\,3^2+5^2\\ =&\,34. \end{align*}

Therefore, the equation of $C$ is

\begin{align*} (x-1)^2+(y-10)^2=34. \end{align*}

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圆心是 $(1,10)$ ，离 $x$ -axis 最近的点在圆心正下方，所以 $x$ 坐标不变， $y$ 坐标减去半径。

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The centre is $(1,10)$ and the radius is $\sqrt{34}$ .

The point on $C$ nearest the $x$ -axis is vertically below the centre.

Therefore, its coordinates are

\begin{align*} (1,10-\sqrt{34}). \end{align*}