📚 Circles | 圆
In A-Level Edexcel Mathematics, circles are a central topic in coordinate geometry. This guide covers standard equations, completing the square, tangents, chords, intersections, and common exam techniques. You will learn how to identify the centre and radius from different forms of the circle equation, how to use perpendicular gradients to find tangents, and how the discriminant helps determine the number of intersection points between a line and a circle.
在A-Level Edexcel数学中，圆是坐标几何的核心主题。本指南涵盖标准方程、配方法、切线、弦、交点以及常见考试技巧。你将学会如何从不同形式的圆方程中识别圆心和半径，如何使用垂直梯度求切线，以及判别式如何帮助确定直线与圆的交点个数。
1. Standard Equation of a Circle | 圆的标准方程
on (x − a)² + (y − b)² = r². If the centre is the origin, this simplifies to x² + y² = r².
以 (a, b) 为圆心、r 为半径的圆的笛卡尔方程为 (x − a)² + (y − b)² = r²。如果圆心在原点，则简化为 x² + y² = r²。
The equation comes directly from Pythagoras' theorem: every point (x, y) on the circle is a fixed distance r from the centre (a, b). This is the most useful form because the centre and radius can be read off immediately.
该方程直接来自勾股定理：圆上的每一点 (x, y) 到圆心 (a, b) 的距离都等于固定长度 r。这是最有用的形式，因为圆心和半径可以直接读出。
| Centre | Radius | Equation |
|---|---|---|
| (a, b) | r | (x − a)² + (y − b)² = r² |
| (0, 0) | r | x² + y² = r² |
2. Completing the Square | 配方法求圆心与半径
lete the square for x and y separately to rewrite it as (x + g)² + (y + f)² = g² + f² − c.
如果圆以展开式给出，例如 x² + y² + 2gx + 2fy + c = 0，请分别对 x 和 y 配平方，将其改写为 (x + g)² + (y + f)² = g² + f² − c。
This reveals the centre (−g, −f) and radius √g² + f² − c. Completing the square is the standard first step when the circle equation is not already in standard form.
由此可得圆心 (−g, −f)，半径 √g² + f² − c。当圆方程不是标准形式时，配平方是标准的第一步。
x² + y² + 2gx + 2fy + c = 0 ⇒ centre (−g, −f), radius = √g² + f² − c
3. General Form and Radius Condition | 一般式与半径条件
The expression g² + f² − c determines whether the equation represents a real circle. If g² + f² − c > 0, there is a real circle; if it equals zero, the circle degenerates to a point; if it is negative, there is no real circle.
表达式 g² + f² − c 决定方程是否表示一个真实圆。如果 g² + f² − c > 0，则存在真实圆；如果等于 0，圆退化为一个点；如果为负，则不存在真实圆。
You should always check this condition when given a general second-degree equation, especially if the radius appears to involve a square root of a negative number.
当给定一般二次方程时，你应始终检查该条件，尤其是当半径似乎涉及负数平方根时。
| Condition | Meaning |
|---|---|
| g² + f² − c > 0 | Real circle with positive radius |
| g² + f² − c = 0 | Point circle (radius 0) |
| g² + f² − c < 0 | No real circle |
4. Points and Circles | 点与圆的位置关系
To determine whether a point P(x₁, y₁) lies inside, on, or outside a circle, compare the distance from the centre to the point with the radius. Alternatively, substitute the point into the circle equation: if the left-hand side is less than r², the point is inside; if equal, on; if greater, outside.
要确定点 P(x₁, y₁) 是在圆内、圆上还是圆外，请比较圆心到该点的距离与半径。或者，将该点代入圆方程：如果左侧小于 r²，则点在圆内；如果等于 r²，则点在圆上；如果大于 r²，则点在圆外。
For (x − a)² + (y − b)² = r², evaluate (x₁ − a)² + (y₁ − b)²
= 25, the point (5, 1) gives (5 − 2)² + (1 + 3)² = 9 + 16 = 25, so it lies on the circle.
示例：对于圆 (x − 2)² + (y + 3)² = 25，点 (5, 1) 代入得 (5 − 2)² + (1 + 3)² = 9 + 16 = 25，因此它在圆上。
5. Tangents to a Circle | 圆的切线
ocal for the tangent gradient, and finally use the point-gradient form y − y₁ = m(x − x₁).
在圆上任意一点，切线垂直于该点处的半径。要找到切线方程，首先计算半径的斜率，然后取负倒数作为切线斜率，最后使用点斜式 y − y₁ = m(x − x₁)。
f y₁ = b, the radius is horizontal and the tangent is vertical, so its equation is x = x₁.
对于圆心为 (a, b)、点 P(x₁, y₁) 在圆上的圆，切线斜率为 m_切线 = −(x₁ − a)/(y₁ − b)，前提是 y₁ ≠ b。如果 y₁ = b，半径是水平的，切线是垂直的，因此其方程为 x = x₁。
mradius = (y₁ − b)/(x₁ − a), m_tangent = − (x₁ − a)/(y₁ − b)
6. Chord Bisector Property | 弦的垂直平分线性质
The perpendicular from the centre of a circle to a chord bisects the chord. Equivalently, the perpendicular bisector of any chord passes through the centre of the circle.
从圆心到弦的垂线平分该弦。等价地，任何弦的垂直平分线都经过圆心。
This property is often used to find the centre when given two chords, or to find the midpoint of a chord when given a line through the centre. In coordinate geometry, if M is the midpoint of a chord AB, then the line from the centre to M is perpendicular to AB.
该性质常用于已知两条弦求圆心，或已知经过圆心的直线求弦的中点。在坐标几何中，如果 M 是弦 AB 的中点，则从圆心到 M 的直线垂直于 AB。
If M is midpoint of chord AB, then CM ⟂ AB, where C is the centre.
7. Intersection of a Line and a Circle | 直线与圆的交点
. This gives a quadratic in one variable. Solve it to find the coordinates of the intersection points.
要求直线与圆的交点，请将直线方程 y = mx + c（或 x = k）代入圆方程。这会得到一个关于一个变量的二次方程。求解该方程即可得到交点坐标。
two distinct points, Δ = 0 means the line is tangent, and Δ < 0 means no intersection.
该二次方程的判别式 Δ = b² − 4ac 可以判断交点个数：Δ > 0 表示两个不同交点，Δ = 0 表示直线为切线，Δ < 0 表示没有交点。
Substitute line into circle → quadratic → Δ = b² − 4ac
8. Length of a Chord | 弦长
If a line cuts a circle, the length of the chord can be found using the perpendicular distance d from the centre to the line. The chord length is 2√r² − d².
如果一条直线与圆相交，弦长可以通过圆心到直线的垂直距离 d 来求。弦长为 2√r² − d²。
This avoids solving the quadratic for both intersection points. Calculate d using the perpendicular distance formula, then apply Pythagoras to the right triangle formed by the radius, half-chord, and perpendicular distance.
这避免了解二次方程求两个交点。使用点到直线的距离公式计算 d，然后在半径、半
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