📚 A-Level Maths: Standard Equation of a Circle and Applications | A-Level 数学：圆的标准方程与应用
In coordinate geometry, the circle is one of the most important shapes because its equation combines algebra, geometry, and distance relationships. Mastering the standard equation of a circle helps you solve problems involving tangents, intersections, and locus conditions, all of which appear regularly in A-Level examinations.
在坐标几何中，圆是最重要的图形之一，因为它的方程将代数、几何与距离关系紧密结合。掌握圆的标准方程，可以帮助你解决涉及切线、交点以及轨迹条件的问题，这些在 A-Level 考试中经常出现。
1. Definition of a Circle | 圆的定义
A circle is defined as the set of all points in a plane that are at a fixed distance, called the radius, from a fixed point, called the centre.
圆的定义是：平面内到一定点距离等于定长的所有点组成的集合。这个定点称为圆心，定长称为半径。
If the centre is at the point (a, b) and the radius is r, then for any point (x, y) on the circle, the distance from (x, y) to (a, b) must equal r.
若圆心为 (a, b)，半径为 r，那么对于圆上任意一点 (x, y)，它到圆心 (a, b) 的距离都等于 r。
(x – a)² + (y – b)² = r²
This is the standard equation of a circle.
这就是圆的标准方程。
2. Standard Form: Centre and Radius | 标准形式：圆心与半径
The standard form of a circle’s equation is immediately informative: you can read the centre and the radius directly from it.
圆的标准方程具有信息直观的特点：你可以直接从中读出圆心和半径。
In the equation (x – a)² + (y – b)² = r², the centre is (a, b) and the radius is r. Note that r > 0 for a real circle.
在方程 (x – a)² + (y – b)² = r² 中，圆心为 (a, b)，半径为 r。注意 r > 0 才表示实际存在的圆。
For example, (x – 3)² + (y + 2)² = 25 represents a circle with centre (3, -2) and radius 5 because 5² = 25. Notice that y + 2 is written as y – (-2), so the y-coordinate of the centre is -2.
例如，(x – 3)² + (y + 2)² = 25 表示圆心为 (3, -2)，半径为 5 的圆，因为 5² = 25。注意 y + 2 可写为 y – (-2)，因此圆心的 y 坐标为 -2。
3. Deriving the Equation from Geometric Conditions | 由几何条件推导方程
You may need to find the equation of a circle from certain geometric conditions, such as knowing the endpoints of a diameter, or the centre and a point on the circle.
有时你需要根据几何条件求圆的方程，例如已知直径的两个端点，或者圆心和圆上一点。
If you know the two endpoints (x₁, y₁) and (x₂, y₂) of a diameter, the centre is their midpoint:
若已知直径的两个端点为 (x₁, y₁) 和 (x₂, y₂)，则圆心是它们的中点：
a = (x₁ + x₂)/2, b = (y₁ + y₂)/2
The radius is half the distance between the endpoints, or equivalently the distance from the midpoint to either endpoint.
半径等于两个端点间距离的一半，也就是中点到一个端点的距离。
Example: endpoints of a diameter are (1, 2) and (5, -4). The centre is (3, -1). The radius is √[(5 – 1)² + (-4 – 2)²]/2 = √52/2 = √13. Thus the equation is (x – 3)² + (y + 1)² = 13.
例：直径端点为 (1, 2) 和 (5, -4)，则圆心为 (3, -1)。半径为 √[(5 – 1)² + (-4 – 2)²]/2 = √52/2 = √13。因此方程为 (x – 3)² + (y + 1)² = 13。
4. General Form and Completing the Square | 一般形式与配方
The general form of a circle’s equation is often written as x² + y² + 2gx + 2fy + c = 0, where g, f and c are constants.
圆的一般形式通常写为 x² + y² + 2gx + 2fy + c = 0，其中 g、f、c 为常数。
To find the centre and radius from this form, you must complete the square for x and y.
要从这种形式求出圆心和半径，需要对 x 和 y 分别配方。
For x: x² + 2gx = (x + g)² – g². For y: y² + 2fy = (y + f)² – f². Substituting back gives:
对 x：x² + 2gx = (x + g)² – g²。对 y：y² + 2fy = (y + f)² – f²。代回后得到：
(x + g)² + (y + f)² = g² + f² – c
Therefore the centre is (-g, -f) and the radius is √(g² + f² – c), provided g² + f² – c > 0.
因此圆心为 (-g, -f)，半径为 √(g² + f² – c)，条件为 g² + f² – c > 0。
5. Finding Centre and Radius from Equation | 从方程求圆心和半径
When you are given an equation like x² + y² – 6x + 4y – 12 = 0, you should always rearrange it into standard form.
当你遇到像 x² + y² – 6x + 4y – 12 = 0 这样的方程时，应始终将其整理为标准形式。
Group x terms together and y terms together:
先将 x 项和 y 项分别分组：
(x² – 6x) + (y² + 4y) = 12
Complete the square for x: (x – 3)² – 9. For y: (y + 2)² – 4. Then the equation becomes:
对 x 配方：(x – 3)² – 9。对 y 配方：(y + 2)² – 4。于是方程变为：
(x – 3)² + (y + 2)² = 25
Hence the centre is (3, -2) and the radius is 5. Always check the constant term transfers correctly to the right-hand side.
因此圆心为 (3, -2)，半径为 5。务必检查常数项是否正确移到等号右侧。
6. Points on, Inside, Outside the Circle | 点在圆上、圆内、圆外的判定
Given the standard equation (x – a)² + (y – b)² = r², you can determine the position of any point (x₁, y₁) relative to the circle.
给定标准方程 (x – a)² + (y – b)² = r²，你可以判断任意点 (x₁, y₁) 与圆的相对位置。
Compute the value (x₁ – a)² + (y₁ – b)² and compare it with r².
计算 (x₁ – a)² + (y₁ – b)² 并和 r² 比较。
- If (x₁ – a)² + (y₁ – b)² = r², the point lies on the circle.
- If (x₁ – a)² + (y₁ – b)² < r², the point lies inside the circle.
- If (x₁ – a)² + (y₁ – b)² > r², the point lies outside the circle.
若 (x₁ – a)² + (y₁ – b)² = r²，则点在圆上；若小于 r²，则在圆内；若大于 r²，则在圆外。
For example, for the circle (x – 2)² + (y – 1)² = 9, the point (5, 1) gives (5 – 2)² + (1 – 1)² = 9, so it lies on the circle. The point (2, 5) gives 0 + 16 = 16 > 9, so it is outside.
例如，对于圆 (x – 2)² + (y – 1)² = 9，点 (5, 1) 的值为 (5 – 2)² + (1 – 1)² = 9，因此点在圆上。点 (2, 5) 的值为 0 + 16 = 16 > 9，因此在圆外。
7. Intersection of Line and Circle | 直线与圆的相交
To find where a straight line intersects a circle, substitute the line’s equation into the circle’s equation and solve the resulting quadratic equation.
要求直线与圆的交点，只需将直线方程代入圆的方程，然后解所得的一元二次方程。
Suppose the line is y = mx + c and the circle is (x – a)² + (y – b)² = r². Replace y in the circle equation, expand, and collect terms into the form Ax² + Bx + C = 0.
设直线为 y = mx + c，圆为 (x – a)² + (y – b)² = r²。将 y 代入圆的方程，展开并整理成 Ax² + Bx + C = 0 的形式。
The discriminant Δ = B² – 4AC tells you the type of intersection:
判别式 Δ = B² – 4AC 决定交点情况：
- Δ > 0: two distinct intersection points.
- Δ = 0: exactly one point, meaning the line is tangent to the circle.
- Δ < 0: no intersection, the line misses the circle.
Δ > 0：有两个不同交点；Δ = 0：恰好一个交点，即切线；Δ < 0：没有交点，直线与圆相离。
8. Tangent to a Circle | 圆的切线
A tangent to a circle is a straight line that touches the circle at exactly one point. The tangent is always perpendicular to the radius at the point of contact.
圆的切线是与圆恰好只有一个公共点的直线。切线与过切点的半径始终垂直。
If the centre is (a, b) and the point of contact is (x₁, y₁), the slope of the radius is (y₁ – b)/(x₁ – a). The tangent slope is the negative reciprocal, provided the radius is not vertical.
若圆心为 (a, b)，切点为 (x₁, y₁)，则半径的斜率为 (y₁ – b)/(x₁ – a)。切线的斜率是该斜率的负倒数（当半径不竖直时）。
A more direct formula: for the circle (x – a)² + (y – b)² = r², the equation of the tangent at (x₁, y₁) is:
更直接的公式：对于圆 (x – a)² + (y – b)² = r²，在点 (x₁, y₁) 处的切线方程为：
(x₁ – a)(x – x₁) + (y₁ – b)(y – y₁) = 0
Or equivalently: (x₁ – a)(x – a) + (y₁ – b)(y – b) = r².
等价地： (x₁ – a)(x – a) + (y₁ – b)(y – b) = r²。
9. Applications: Circle Geometry Problems | 应用：圆几何问题
Circle equations can be used to solve many practical coordinate geometry problems. For example, you may be asked to find the equation of a circle that passes through three given non-collinear points.
圆的方程可用于解决许多坐标几何实际问题。例如，你可能会被要求求经过三个不共线点的圆的方程。
One method is to substitute each point into the general form x² + y² + 2gx + 2fy + c = 0, producing three simultaneous equations in g, f and c. Solve them to find the centre and radius.
一种方法是将每个点代入一般形式 x² + y² + 2gx + 2fy + c = 0，得到关于 g、f、c 的三个联立方程，解方程即可求得圆心和半径。
Another common application is finding the locus of a moving point. For example, if a point P moves so that its distance from a fixed point A is always equal to 5, then the locus of P is a circle with centre A and radius 5.
另一个常见应用是求动点的轨迹。例如，若动点 P 到定点 A 的距离始终等于 5，则 P 的轨迹是以 A 为圆心、半径为 5 的圆。
Tangents and normals to circles also appear in problems about shortest distances from a point to a circle, or finding the equation of a line that just touches a circle with a given slope.
圆的切线与法线也常出现在求点到圆的最短距离，或求具有给定斜率且与圆相切的直线方程等问题中。
10. Practice Exam-Style Questions | 考试型练习
Let’s look at a typical A-Level style question: A circle has centre C(2, -3) and passes through the point P(6, 0). Find the equation of the circle and the equation of the tangent at P.
我们来看一道典型的 A-Level 风格题目：已知圆圆心为 C(2, -3)，且经过点 P(6, 0)。求圆的方程以及点 P 处切线的方程。
The radius is the distance from C to P:
半径为 C 到 P 的距离：
r = √[(6 – 2)² + (0 – (-3))²] = √(16 + 9) = 5
Thus the circle equation is (x – 2)² + (y + 3)² = 25.
因此圆的方程为 (x – 2)² + (y + 3)² = 25。
At P(6, 0), the radius CP has slope (0 + 3)/(6 – 2) = 3/4. The tangent slope is -4/3. Using y = mx + c, since the tangent passes through P:
在 P(6, 0)，半径 CP 的斜率为 (0 + 3)/(6 – 2) = 3/4。切线斜率为 -4/3。利用 y = mx + c，切线过点 P：
0 = (-4/3)(6) + c ⇒ c = 8
So the tangent equation is y = -4x/3 + 8, or 4x + 3y – 24 = 0.
所以切线方程为 y = -4x/3 + 8，即 4x + 3y – 24 = 0。
11. Common Mistakes and Tips | 常见错误与技巧
A frequent mistake is misreading the centre from an equation like (x + 2)² + (y – 3)² = 4. The centre is (-2, 3), not (2, -3), because the signs are opposite to the brackets.
一个常见错误是误读圆心的坐标，例如 (x + 2)² + (y – 3)² = 4 中，圆心是 (-2, 3)，而不是 (2, -3)，因为括号内符号相反。
Another common error is forgetting that the right-hand side is r², not r. If the equation says 16, the radius is 4, not 16.
另一个常见错误是忘记等号右侧是 r² 而不是 r。若右侧为 16，则半径为 4，而不是 16。
When using the general form, always complete the square carefully. Check that the coefficients of x² and y² are both 1. If they are not, divide the whole equation by the coefficient first.
使用一般形式时，务必仔细配方。先检查 x² 和 y² 的系数是否都为 1。如果不是，先将整个方程除以该系数。
When testing intersection of a line and circle, write the quadratic in the form Ax² + Bx + C = 0 before calculating the discriminant. A missing constant term leads to wrong conclusions.
判断直线与圆相交时，先将二次方程写成 Ax² + Bx + C = 0 的形式，再计算判别式。常数项遗漏会导致错误结论。
12. Summary | 总结
The standard equation (x – a)² + (y – b)² = r² is the foundation of circle geometry in coordinate systems. From it you can quickly identify the centre and radius, test whether points lie on or off the circle, and find tangents or intersections.
标准方程 (x – a)² + (y – b)² = r² 是坐标系中圆几何的基础。由它可以快速识别圆心和半径，判断点与圆的位置关系，并求切线或交点。
You should be able to convert between standard form and general form using completing the square, and to apply discriminant analysis to lines and circles.
你应该能够通过配方在标准形式与一般形式之间转换，并运用判别式分析直线与圆的关系。
Practising these skills will prepare you well for both pure mathematics and coordinate geometry questions in A-Level exams.
充分练习这些技能，将帮助你在 A-Level 考试中从容应对纯数学与坐标几何题目。
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