📚 Mastering Coordinate Geometry for A-Level Maths | A-Level 数学：坐标几何 考点精讲
Coordinate geometry, also known as analytic geometry, is a vital component of the A-Level Mathematics syllabus. It bridges algebra and geometry by representing geometric shapes with equations and using algebraic techniques to solve spatial problems. In this revision guide, we will walk through every essential concept: from line equations and circles to parametric curves and tangents. Whether you are preparing for Paper 1 or tackling mixed problems, these structured notes will sharpen your understanding and exam technique.
坐标几何，也称解析几何，是A-Level数学大纲的重要组成部分。它通过方程表示几何图形，并利用代数方法解决空间问题，从而将代数与几何联系起来。在本考点精讲中，我们将梳理每一个核心概念：从直线方程、圆到参数曲线与切线。无论你正在备考Paper 1还是应对综合题，这些结构化的笔记都将帮助你深化理解、提升应试技巧。
1. Basic Concepts: Distance and Midpoint | 基础概念：距离与中点
The distance between two points A(x₁, y₁) and B(x₂, y₂) is given by the formula derived from Pythagoras' theorem.
两点A(x₁, y₁)与B(x₂, y₂)之间的距离公式由勾股定理推导而来。
d = √((x₂ − x₁)² + (y₂ − y₁)²). This is used extensively to find lengths of line segments.
距离公式 d = √((x₂ − x₁)² + (y₂ − y₁)²)。它被广泛用于求线段长度。
The midpoint M of AB has coordinates ((x₁ + x₂)/2, (y₁ + y₂)/2). It is the average of the endpoints.
线段AB的中点M坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)，即端点坐标的平均值。
Always ensure you substitute coordinates carefully, especially when negatives appear. These formulas form the building blocks for many proofs and constructions.
代入坐标时务必小心，尤其是出现负数时。这些公式是许多证明与作图的基础。
2. Gradient of a Straight Line | 直线的斜率
For two distinct points (x₁, y₁) and (x₂, y₂), the gradient is m = (y₂ − y₁) / (x₂ − x₁).
斜率 m 衡量一条线的倾斜程度。对于两个不同的点 (x₁, y₁) 和 (x₂, y₂)，斜率为 m = (y₂ − y₁) / (x₂ − x₁)。
ns it falls. A horizontal line has m = 0, while a vertical line has an undefined gradient.
斜率为正表示直线从左到右上升；斜率为负表示下降。水平线的斜率为0，垂直线的斜率无定义。
If three points are collinear, the gradients between any two pairs are equal. This is a useful check for alignment.
若三点共线，任意两点间的斜率相等。这是检验共线性的有用方法。
Gradient is also the tangent of the angle the line makes with the positive x‑axis: m = tan θ.
斜率也等于直线与x轴正方向夹角的正切：m = tan θ。
3. Equations of Straight Lines | 直线方程
equation are: y = mx + c (slope-intercept form) and y − y₁ = m(x − x₁) (point-slope form).
最常见的直线方程形式有：y = mx + c（斜截式）和 y − y₁ = m(x − x₁)（点斜式）。
Here c is the y‑intercept, where the line crosses the y‑axis (x = 0). In the point-slope form, (x₁, y₁) is a known point on the line.
其中 c 是 y 轴截距，即直线与 y 轴（x = 0）的交点。点斜式中 (x₁, y₁) 为直线上一个已知点。
ally useful for finding intersections and distances. You can convert between forms easily.
一般式 ax + by + c = 0 在求交点与距离时特别有用。不同形式之间可以轻松转换。
To find the equation of a line through two points, first compute the gradient, then use point-slope form with either point.
求经过两点的直线方程时，先计算斜率，再选用任一点代入点斜式。
4. Parallel and Perpendicular Lines | 平行与垂直条件
rallel if and only if their gradients are equal: m₁ = m₂. Their equations have identical x‑ and y‑coefficients ratios.
两条不同的直线平行当且仅当它们的斜率相等：m₁ = m₂。它们的方程中 x 与 y 系数之比相同。
the product of their gradients is −1: m₁ × m₂ = −1. This is the negative reciprocal rule.
两条直线垂直的条件是斜率之积为 −1：m₁ × m₂ = −1。即负倒数规则。
For example, if a line has gradient 3, a line perpendicular to it will have gradient −1/3. Vertical and horizontal lines are also perpendicular, though one gradient is undefined.
例如，若一条直线斜率为 3，与之垂直的直线斜率为 −1/3。竖直线与水平线也互相垂直，尽管竖直线斜率无定义。
These relationships are key for finding altitudes of triangles, normals to curves, and constructing rectangles in coordinate geometry problems.
这些关系是求三角形的高、曲线的法线以及在坐标几何题中构造矩形的关键。
5. Circles: Standard and General Form | 圆的方程：标准式与一般式
standard equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r².
以 (a, b) 为圆心、r 为半径的圆的标准方程为 (x − a)² + (y − b)² = r²。
e general form x² + y² + 2gx + 2fy + c = 0, where the centre is (−g, −f) and the radius is √g² + f² − c, provided g² + f² − c > 0.
将标准式展开可得一般式 x² + y² + 2gx + 2fy + c = 0，其中圆心为 (−g, −f)，半径为 √g² + f² − c，且须满足 g² + f² − c > 0。
f² − c = 0, the equation represents a point circle; if negative, there is no real circle.
若 g² + f² − c = 0，方程表示一个点圆；若为负，则表示不存在实圆。
Recognising the centre and radius from either form is essential for sketching and solving geometric problems.
从任一形式识别圆心与半径对于绘图和求解几何问题至关重要。
6. Finding Centre and Radius by Completing the Square | 配方法求圆心和半径
= 0, you must complete the square for both x and y terms to reveal its centre and radius.
当给出的圆方程为 x² + y² + Dx + Ey + F = 0 时，必须对 x 项和 y 项分别配方，才能得出圆心和半径。
Group x² + Dx as (x + D/2)² − (D/2)², and similarly for y. Then rearrange into standard form.
将 x² + Dx 组合成 (x + D/2)² − (D/2)²，y 项同理。然后移项整理成标准式。
y² − 6x + 2y − 6 = 0 becomes (x − 3)² + (y + 1)² = 16, giving centre (3, −1) and radius 4.
例题：x² + y² − 6x + 2y − 6 = 0 可化为 (x − 3)² + (y + 1)² = 16，故圆心为 (3, −1)，半径为 4。
This technique is frequently tested; always check the sign of the constant term after moving it to the right-hand side.
配方法常被考查；将常数项移至等式右侧后务必核对符号。
7. Intersection of a Line and a Circle | 直线与圆的交点
lly y = mx + c or x = k) into the circle equation, generating a quadratic in one variable.
要求直线与圆的交点，可将直线方程（一般为 y = mx + c 或 x = k）代入圆的方程，得到关于一个变量的二次方程。
secant), Δ = 0 gives one intersection (line is tangent), and Δ < 0 means no intersection.
该二次方程的判别式 Δ = b² − 4ac 决定交点个数：Δ > 0 表示有两个交点（直线为割线），Δ = 0 表示一个交点（切线），Δ < 0 表示无交点。
exactly one point, and you may need to find that point by solving the equation when Δ = 0.
求解时一定要回归题意：切线与圆恰好相切于一点，你可能需要在 Δ = 0 时解方程求出该切点。
This discriminant method avoids lengthy geometric constructions and is highly algebraic, suiting A‑Level exam styles.
判别式法避免了冗长的几何作图，高度代数化，契合A‑Level考试风格。
8. Tangents and Normals to a Circle | 圆的切线与法线
A tangent to a circle at a given point P(x₁, y₁) on the circle is perpendicular to the radius at P. The gradient of the radius from centre (a, b) is (y₁ − b)/(x₁ − a), so the tangent's gradient is the negative reciprocal.
圆上给定点 P(x₁, y₁) 处的切线垂直于该点处的半径。从圆心 (a, b) 到 P 的半径斜率为 (y₁ − b)/(x₁ − a)，因此切线的斜率为其负倒数。
x₁ − a)(x − x₁) + (y₁ − b)(y − y₁) = 0, or using point-slope form with the found gradient.
因此切线方程可写为 (x₁ − a)(x − x₁) + (y₁ − b)(y − y₁) = 0，或利用求得的斜率代入点斜式。
The normal at P is the line through P perpendicular to the tangent, i.e., along the radius. Its equation is easy to derive once the centre is known.
点 P 处的法线是经过 P 且垂直于切线的直线，即沿着半径的方向。一旦知道圆心，法线方程便容易求得。
o the circle, impose Δ = 0, and solve for the gradient. This yields two possible tangents.
对于从圆外一点 Q 引出的切线，可设含未知斜率的直线方程，代入圆的方程，令 Δ = 0 以解出斜率。这样可得到两条可能的切线。
9. Parametric Equations | 参数方程简介
ommon parametrisation is x = a + r cos t, y = b + r sin t, where t is the angle parameter.
坐标几何中的许多曲线用第三个变量——参数——来表示。对圆而言，常用的参数方程为 x = a + r cos t, y = b + r sin t，其中 t 为角度参数。
iminate the parameter using algebraic manipulation or identities like cos² t + sin² t = 1.
其他重要的参数曲线包括抛物线（x = t², y = 2t）和椭圆。要转化为笛卡儿形式，需通过代数运算或利用恒等式（如 cos² t + sin² t = 1）消去参数。
Calculus can be applied to parametric curves: the gradient dy/dx is given by (dy/dt) / (dx/dt), provided dx/dt ≠ 0. This links coordinate geometry with differentiation.
微积分可应用于参数曲线：导数 dy/dx 由 (dy/dt) / (dx/dt) 给出，前提是 dx/dt ≠ 0。这就把坐标几何与微分学联系了起来。
In exams, you might be asked to find equations of tangents and normals to parametric curves at specific parameter values.
考试中可能会要求求参数曲线在特定参数值处的切线和法线方程。
10. Problem-solving Strategies | 解题策略
Always begin by sketching the geometry or at least noting key features: intercepts, centres, and gradients. A rough diagram can prevent sign errors.
解题总要从画出几何草图或至少标注关键特征开始：截距、圆心和斜率。一幅粗略的草图可避免符号错误。
stance formula; for perpendicularity, use m₁m₂ = −1; for tangency, set Δ = 0 or use radius‑perpendicular property.
若题目涉及长度，回忆距离公式；涉及垂直关系，使用 m₁m₂ = −1；涉及相切，可设 Δ = 0 或利用半径垂直于切线的性质。
Many exam questions combine a circle with a triangle formed by a chord and radii. Use midpoint of the chord and perpendicular bisector properties.
许多考题将圆与由弦和半径构成的三角形相结合。要利用弦中点和垂直平分线的性质。
For locus problems, express the condition algebraically (e.g., distance from a fixed point is constant) and simplify to an equation in x and y. This often results in a circle or a straight line.
对于轨迹问题，将条件代数化（例如到定点的距离为常数），并化简为关于 x 和 y 的方程。结果通常是一个圆或一条直线。
Coordinate geometry regularly appears in synoptic questions, linking to surds, quadratics, and calculus. Practise mixed exercises to build fluency.
坐标几何经常出现在综合性试题中，与根式、二次方程和微积分结合。通过混合练习提升熟练度。
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