📚 A-Level Maths: Standard and General Form of a Circle | 圆的标准方程与一般方程
A circle is one of the most important objects in coordinate geometry. In A-Level mathematics, you need to know two equivalent forms of its equation: the standard form and the general form. This lesson explains how to use both forms, how to convert between them, and how to read off the centre and radius quickly.
圆是坐标几何中最重要的研究对象之一。在 A-Level 数学中，你需要掌握圆的两种等价方程形式：标准方程与一般方程。本节课将讲解这两种形式的使用方法、它们之间的互化，以及如何快速求出圆心和半径。
1. The Standard Form of a Circle | 圆的标准方程
The standard form of a circle with centre (a, b) and radius r is:
(x – a)² + (y – b)² = r²
This form is useful because you can read the centre and radius directly. For example, (x – 1)² + (y + 2)² = 9 has centre (1, -2) and radius 3.
圆心为 (a, b)、半径为 r 的圆的标准方程为：
(x – a)² + (y – b)² = r²
该形式的优点在于可以直接读出圆心和半径。例如，(x – 1)² + (y + 2)² = 9 的圆心为 (1, -2)，半径为 3。
If the centre is at the origin (0, 0), the equation simplifies to x² + y² = r².
若圆心在原点 (0, 0)，方程简化为 x² + y² = r²。
2. The General Form of a Circle | 圆的一般方程
By expanding the standard form and rearranging, we obtain the general form:
x² + y² + Dx + Ey + F = 0
where D, E and F are constants. A valid circle equation must have equal coefficients for x² and y², and must not contain an xy term.
将标准方程展开并整理，可以得到圆的一般方程：
x² + y² + Dx + Ey + F = 0
其中 D、E、F 为常数。有效的圆方程必须满足 x² 与 y² 的系数相等，并且不含 xy 项。
If the coefficients of x² and y² are not equal, or an xy term appears, the equation describes a different conic such as an ellipse or a rotated curve.
如果 x² 与 y² 的系数不相等，或者出现了 xy 项，那么该方程描述的是其他圆锥曲线，例如椭圆或旋转后的曲线。
3. Converting Standard Form to General Form | 从标准形式到一般形式
Starting from (x – a)² + (y – b)² = r², expand the two brackets:
x² – 2ax + a² + y² – 2by + b² = r²
Move r² to the left-hand side:
x² + y² – 2ax – 2by + (a² + b² – r²) = 0
Comparing with x² + y² + Dx + Ey + F = 0 gives D = -2a, E = -2b and F = a² + b² – r².
从 (x – a)² + (y – b)² = r² 出发，展开两个括号：
x² – 2ax + a² + y² – 2by + b² = r²
将 r² 移到等号左边：
x² + y² – 2ax – 2by + (a² + b² – r²) = 0
与 x² + y² + Dx + Ey + F = 0 对比可得 D = -2a，E = -2b，F = a² + b² – r²。
4. Converting General Form to Standard Form | 从一般形式到标准形式
To convert x² + y² + Dx + Ey + F = 0 into standard form, group the x terms and y terms, then complete the square for each group.
将 x² + y² + Dx + Ey + F = 0 化为标准形式，需要先将 x 项和 y 项分别分组，然后分别配方。
Completing the square gives:
(x + D/2)² – (D/2)² + (y + E/2)² – (E/2)² + F = 0
Rearrange the constants to the right-hand side:
(x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² – F
For example, convert x² + y² + 6x – 4y – 3 = 0.
例如，将 x² + y² + 6x – 4y – 3 = 0 化为标准形式。
(x + 3)² – 9 + (y – 2)² – 4 – 3 = 0
(x + 3)² + (y – 2)² = 16
So the centre is (-3, 2) and the radius is 4.
因此圆心为 (-3, 2)，半径为 4。
5. Finding the Centre and Radius | 求圆心与半径
When given the general form x² + y² + Dx + Ey + F = 0, you can find the centre and radius using these results:
当给出一般方程 x² + y² + Dx + Ey + F = 0 时，可以利用以下结论求圆心和半径：
- Centre = (-D/2, -E/2)
- 圆心为 (-D/2, -E/2)
- Radius r = √[(D/2)² + (E/2)² – F]
- 半径 r = √[(D/2)² + (E/2)² – F]
This method is fast and avoids writing out the full completing-square process each time.
这种方法非常快捷，可以避免每次重新书写完整的配方过程。
6. Condition for a Real Circle | 一般方程表示实圆的条件
For x² + y² + Dx + Ey + F = 0 to represent a real circle, the radius squared must be positive:
要使 x² + y² + Dx + Ey + F = 0 表示一个实圆，半径的平方必须为正：
r² = (D/2)² + (E/2)² – F > 0
If r² = 0, the circle degenerates into a single point. If r² < 0, no real points satisfy the equation, so the graph does not exist on the real coordinate plane.
若 r² = 0，圆退化成一个点；若 r² < 0，则没有任何实数点满足该方程，因此图像在实坐标平面内不存在。
7. Worked Example 1: Standard Form to General Form | 例题一：标准形式化一般形式
Write the equation (x – 2)² + (y + 3)² = 16 in general form.
将方程 (x – 2)² + (y + 3)² = 16 写成一般形式。
Expand the brackets:
x² – 4x + 4 + y² + 6y + 9 = 16
Simplify:
x² + y² – 4x + 6y – 3 = 0
Therefore, the general form is x² + y² – 4x + 6y – 3 = 0.
因此，一般形式为 x² + y² – 4x + 6y – 3 = 0。
8. Worked Example 2: General Form to Standard Form | 例题二：一般形式化标准形式
A circle has equation x² + y² + 10x – 8y + 5 = 0. Find its centre and radius.
已知圆的方程为 x² + y² + 10x – 8y + 5 = 0，求其圆心和半径。
Group the x terms and y terms:
(x² + 10x) + (y² – 8y) + 5 = 0
Complete the square for each group:
(x + 5)² – 25 + (y – 4)² – 16 + 5 = 0
Simplify:
(x + 5)² + (y – 4)² = 36
Hence the centre is (-5, 4) and the radius is 6.
因此圆心为 (-5, 4)，半径为 6。
9. Worked Example 3: Using the Circle Equation | 例题三：根据条件求圆的方程
A circle has centre (3, -1) and passes through the point (7, 2). Find its equation in general form.
已知圆的圆心为 (3, -1)，且经过点 (7, 2)，求该圆的一般方程。
First find the radius using the distance from the centre to the point:
r = √[(7 – 3)² + (2 – (-1))²] = √(16 + 9) = 5
The standard form is:
(x – 3)² + (y + 1)² = 25
Expand:
x² – 6x + 9 + y² + 2y + 1 = 25
Simplify to obtain the general form:
x² + y² – 6x + 2y – 15 = 0
So the general form is x² + y² – 6x + 2y – 15 = 0.
因此一般方程为 x² + y² – 6x + 2y – 15 = 0。
10. Common Mistakes | 常见错误
Students often make sign errors and arithmetic errors when working with circle equations. Watch for the following points:
学生在处理圆的方程时经常出现符号错误和计算错误，请注意以下几点：
- Remember that (y + 2)² means the y-coordinate of the centre is -2, not +2.
- 注意 (y + 2)² 表示圆心的纵坐标为 -2，而不是 +2。
- When reading the centre from the general form, do not forget to halve D and E.
- 从一般形式读取圆心时，不要忘记将 D 和 E 除以 2。
- When completing the square, add and subtract the same constant so that the equation remains balanced.
- 配方时，必须同时加上和减去同一个常数，以保证方程平衡。
- Always check the condition (D/2)² + (E/2)² – F > 0 before concluding that a real circle exists.
- 在判断是否存在实圆时，务必检查条件 (D/2)² + (E/2)² – F > 0 是否成立。
11. Practice Questions | 练习
Try the following questions before checking the answers.
请先尝试完成以下练习，再对照答案。
| Question | 题目 | Answer | 答案 |
|---|---|
| Write the standard form of the circle with centre (-2, 5) and radius √7. | (x + 2)² + (y – 5)² = 7 |
| Convert x² + y² – 6x + 4y + 4 = 0 to standard form and state the centre and radius. | (x – 3)² + (y + 2)² = 9; centre (3, -2), radius 3 |
| Does x² + y² + 2x – 4y + 10 = 0 represent a real circle? | No, because (x + 1)² + (y – 2)² = -5, so r² < 0. |
Write out your full working, especially when completing the square. This is the best way to avoid small sign errors.
请写出完整的计算过程，尤其是在配方时。这是避免符号错误的最佳方法。
12. Key Points | 要点总结
- The standard form of a circle is (x – a)² + (y – b)² = r², with centre (a, b) and radius r
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