📚 Equation of a Circle | 圆的方程
In coordinate geometry, the equation of a circle is one of the most heavily examined topics because it links three ideas you already know: the distance formula, Pythagoras' theorem and completing the square. A circle is defined as the set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the centre). Once that definition is written algebraically, every question about centres, radii, tangents, chords and intersections becomes a matter of algebra rather than guesswork.
在坐标几何中，圆的方程是考试出现频率极高的内容，因为它把三个你已经掌握的工具连在一起：距离公式、勾股定理与配方法。圆定义为平面上到某个定点（圆心）的距离等于定长（半径）的所有点的集合。一旦把这一几何定义写成代数式，所有关于圆心、半径、切线、弦和交点的问题就都转化为代数运算，而不再是靠猜。
This revision guide covers the standard form, the general form, how to find a circle's equation from geometric conditions, how circles meet lines and other circles, and the mistakes that cost marks most often in A-Level papers.
本篇复习指南将系统讲解标准式、一般式、如何由几何条件求圆的方程、圆与直线及其他圆的相交关系，以及 A-Level 考试中最常见的失分点。
1. Standard Equation of a Circle | 圆的标准方程
If a circle has centre (a, b) and radius r, then every point (x, y) on the circle is exactly r units from the centre. Applying the distance formula and squaring both sides gives the standard equation shown below.
若圆的圆心为 (a, b)、半径为 r，则圆上任意一点 (x, y) 到圆心的距离恰好为 r 个单位。套用距离公式并将两边平方，即得下面的标准方程。
(x − a)² + (y − b)² = r²
ght-hand side is r² and not r. If you are given r = 7, the equation ends in = 49, not = 7.
推导只有一步：√[(x − a)² + (y − b)²] = r，两边平方即可去掉根号。这也解释了为什么等号右边写的是 r² 而不是 r。若已知 r = 7，方程右边应写 49，而不是 7。
implified to y + 1, and 5² is written as 25 immediately so that the radius can be read off later.
例如，圆心为 (2, −1)、半径为 5 的圆，其方程为 (x − 2)² + (y + 1)² = 25。注意 y − (−1) 应化简为 y + 1，同时把 5² 直接写成 25，方便后续读出半径。
A special case worth memorising is the circle centred at the origin, where a = b = 0:
一个必须牢记的特殊情形是圆心在原点，即 a = b = 0 时：
x² + y² = r²
2. Identifying the Centre and Radius | 确定圆心与半径
quare-rooting the right-hand side. In (x + 3)² + (y − 2)² = 16, rewriting x + 3 as x − (−3) shows that the centre is (−3, 2), and since 16 = 4², the radius is 4.
当方程已经是标准式时，括号内数字的相反数就是圆心坐标，等号右边开平方就是半径。对于 (x + 3)² + (y − 2)² = 16，把 x + 3 改写为 x − (−3) 可知圆心为 (−3, 2)；又因为 16 = 4²，所以半径为 4。
(y − 0)² = 10 is simply x² + y² = 10, so the centre is the origin (0, 0) and the radius is √10, which is about 3.16. Zero shifts simplify many problems, but they also cause careless errors when students assume a number must be hiding there.
如果某个括号完全不存在，则圆心对应的坐标为 0。方程 (x − 0)² + (y − 0)² = 10 其实就是 x² + y² = 10，圆心为原点 (0, 0)，半径是 √10，约等于 3.16。零平移会简化很多问题，但学生也常因此粗心，误以为那里还藏着一个数字。
The table below summarises the pattern that examiners expect you to apply instantly.
下表总结了需要你一眼看出的规律。
| Equation | 方程 | Centre | 圆心 | Radius | 半径 |
|---|---|---|
| (x − 4)² + (y − 7)² = 9 | (4, 7) | 3 |
| (x + 5)² + (y + 1)² = 20 | (−5, −1) | √20 = 2√5 |
| x² + (y − 6)² = 1 | (0, 6) | 1 |
| (x + 2)² + y² = ¼ | (−2, 0) | ½ |
Always simplify surds where possible, and never leave a radius as a negative number. A radius is a length, so r > 0.
根式能化简的一定要化简，并且半径永远不能写成负数。半径是长度，所以 r > 0。
3. Deriving the Equation from the Distance Formula | 由距离公式推导圆的方程
and the fixed distance as r, the definition of a circle states that d = r for every point on the curve.
两点 (x₁, y₁) 与 (x₂, y₂) 之间的距离为 d = √[(x₂ − x₁)² + (y₂ − y₁)²]。把定点取为 (a, b)、动点取为 (x, y)、定长取为 r，圆的定义就意味着圆上每一点都满足 d = r。
√[(x − a)² + (y − b)²] = r
Squaring both sides is legitimate here because both sides are non-negative, which gives the standard form again. This derivation is a common 2 or 3 mark question in its own right, so it is worth being able to reproduce it from memory without hesitating.
这里两边平方是合法的，因为两边都非负，平方后再次得到标准式。这段推导本身就是常见的 2 至 3 分小题，因此值得练到能毫不犹豫地默写出来。
e x² + y² = 1, which later reappears in trigonometry as the definition of sine and cosine.
当圆心在原点时，推导简化为 x² + y² = r²。取 r = 1 得到单位圆 x² + y² = 1，它在后续三角函数中会再次出现，成为正弦与余弦的定义基础。
4. The General Form and Completing the Square | 一般式与配方法
Expanding the standard form produces the general form of a circle, usually written with letters D, E and F:
把标准式展开就得到圆的一般式，通常用字母 D、E、F 表示：
x² + y² + Dx + Ey + F = 0
+ (y² + Ey) = −F, and adding the missing squares to both sides produces the result below.
要把一般式还原为标准式，需要对 x 和 y 分别配方。整理可得 (x² + Dx) + (y² + Ey) = −F，两边同时补上缺少的平方项，得到下面的结果。
(x + D/2)² + (y + E/2)² = (D² + E² − 4F) / 4
Therefore the centre is (−D/2, −E/2) and the radius is r = ½√D² + E² − 4F. You may either memorise these two formulas or re-derive them by completing the square each time; the second approach is safer because it also earns method marks.
因此圆心为 (−D/2, −E/2)，半径为 r = ½√D² + E² − 4F。你既可以背下这两个公式，也可以每次重新配方；后者更稳妥，因为过程本身也能拿到方法分。
radius is 6. Notice that the constants −9, −16 and −11 all move to the right-hand side and must be added correctly.
示例：对方程 x² + y² − 6x + 8y − 11 = 0 配方，得 (x − 3)² − 9 + (y + 4)² − 16 − 11 = 0，即 (x − 3)² + (y + 4)² = 36。圆心为 (3, −4)，半径为 6。注意常数 −9、−16 与 −11 都要移到等号右边，符号容易出错。
very term by 2 first to obtain x² + y² − 4x + 6y − 3 = 0, which gives (x − 2)² + (y + 3)² = 16, so the centre is (2, −3) and the radius is 4.
一个重要提醒：使用一般式前，x² 与 y² 的系数必须都等于 1。若给出 2x² + 2y² − 8x + 12y − 6 = 0，应先把每一项除以 2，得到 x² + y² − 4x + 6y − 3 = 0，再配方得 (x − 2)² + (y + 3)² = 16，圆心为 (2, −3)，半径为 4。
5. Conditions and Degenerate Cases | 成立条件与退化情形
cle. After completing the square, the right-hand side equals (D² + E² − 4F) / 4, and the sign of this quantity decides the nature of the locus.
并非所有形如 x² + y² + Dx + Ey + F = 0 的方程都表示一个真正的圆。配方后，等号右边等于 (D² + E² − 4F) / 4，这个量的符号决定了轨迹的性质。
- If D² + E
广告 / Sponsored现货 Cambridge Biology for the IB Diploma coursebook 剑桥国际文凭 I…生物 备考教辅 · 正版现货 · 与本文内容相关查看详情 / View on Taobao参考价 ¥546.0
更多咨询请联系16621398022（同微信）
备考资讯每日更新，关注不错过考纲变化 · 科目分析 · 学习方法
在微信里长按二维码识别关注
或截图后用「扫一扫 → 相册」选择本图公众号：tutorhao