题目
Problem
A circle has centre .
Given that the point lies on ,
(a) (i) find the radius of
(ii) find an equation for
(b) Find the equation of the tangent to at giving your answer in the form where , and are integers to be found.
(7)
题目中文翻译
圆 的圆心为 ，点 在 上。
(a) (i) 求 的半径 (ii) 求 的方程
(b) 求 在点 处的切线方程，写成 的形式，其中 为整数。
（7分）
解答
解法一
思路
半径 = 圆心到圆上一点的距离。圆的标准方程：。切线⊥半径，先求 斜率，取负倒数得切线斜率，再用点斜式。
答题过程
(a)(i) The radius is the distance from the centre to :
r =&\, \sqrt{(8 - 2)^2 + (-3 - 5)^2} \\[4mm] =&\, \sqrt{36 + 64} \\[4mm] =&\, \sqrt{100} \\[4mm] =&\, 10 \end{align*}$$ $$\boxed{r = 10}$$ **(a)(ii)** The equation of circle $C$: $$\boxed{(x - 2)^2 + (y - 5)^2 = 100}$$ **(b)** The gradient of $CP$: $$m_{CP} = \frac{-3 - 5}{8 - 2} = \frac{-8}{6} = -\frac{4}{3}$$ Since the tangent at $P$ is perpendicular to $CP$: $$m_{\text{tangent}} = \frac{3}{4}$$ Using point–slope form at $P(8, -3)$: $$\begin{align*} y - (-3) =&\, \frac{3}{4}(x - 8) \\[4mm] 4(y + 3) =&\, 3(x - 8) \\[4mm] 4y + 12 =&\, 3x - 24 \\[4mm] 3x - 4y - 36 =&\, 0 \end{align*}$$ $$\boxed{3x - 4y - 36 = 0}$$ ## 解法二 #### 思路 隐函数求导法。对圆的方程两边关于 $x$ 求导，代入 $P$ 点坐标求切线斜率，再用点斜式。 #### 答题过程 **(b)** The circle equation is $(x - 2)^2 + (y - 5)^2 = 100$. Differentiating implicitly with respect to $x$: $$2(x - 2) + 2(y - 5)\frac{\mathrm{d}y}{\mathrm{d}x} = 0$$ $$\frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{x - 2}{y - 5}$$ At $P(8, -3)$: $$\frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{8 - 2}{-3 - 5} = -\frac{6}{-8} = \frac{3}{4}$$ Using point–slope form at $P(8, -3)$: $$y + 3 = \frac{3}{4}(x - 8)$$ $$\boxed{3x - 4y - 36 = 0}$$ ## 解法三 #### 思路 切线公式法。圆 $(x-a)^2+(y-b)^2=r^2$ 上点 $(x_1,y_1)$ 处的切线方程为 $(x_1-a)(x-a)+(y_1-b)(y-b)=r^2$，直接代入即可。 #### 答题过程 **(b)** For the circle $(x - 2)^2 + (y - 5)^2 = 100$, the tangent at $P(x_1, y_1)$ is: $$(x_1 - 2)(x - 2) + (y_1 - 5)(y - 5) = 100$$ At $P(8, -3)$: $$\begin{align*} (8 - 2)(x - 2) + (-3 - 5)(y - 5) =&\, 100 \\[4mm] 6(x - 2) + (-8)(y - 5) =&\, 100 \\[4mm] 6x - 12 - 8y + 40 =&\, 100 \\[4mm] 6x - 8y + 28 =&\, 100 \\[4mm] 6x - 8y - 72 =&\, 0 \\[4mm] 3x - 4y - 36 =&\, 0 \end{align*}$$ $$\boxed{3x - 4y - 36 = 0}$$