📚 A-Level WJEC Maths: Coordinate Geometry Essential Revision | A-Level WJEC 数学：坐标几何 考点精讲
Coordinate geometry is a core component of the A-Level WJEC Mathematics specification, bridging algebra and geometry through the Cartesian plane. This guide covers all essential concepts, from straight lines to circles, parametric equations, and loci, with clear explanations and WJEC-style exam insights.
坐标几何是 A-Level WJEC 数学大纲的核心内容，通过笛卡尔平面将代数与几何联系起来。本指南涵盖从直线到圆、参数方程和轨迹的所有基本概念，并提供清晰的解释和 WJEC 风格的考试技巧。
1. Straight Line Basics | 直线基础
A straight line in the coordinate plane can be described using its gradient (slope) and intercepts. The gradient m measures steepness and is calculated as the change in y divided by the change in x between two distinct points.
坐标平面中的直线可以用其梯度（斜率）和截距来描述。梯度 m 用于衡量倾斜程度，计算方式为两点间 y 的变化除以 x 的变化。
If a line passes through points (x₁, y₁) and (x₂, y₂), the gradient is given by:
如果直线通过点 (x₁, y₁) 和 (x₂, y₂)，梯度为：
m = (y₂ − y₁) / (x₂ − x₁), x₂ ≠ x₁
A positive gradient means the line rises from left to right; a negative gradient means it falls. A horizontal line has gradient 0, and a vertical line has an undefined gradient.
正梯度表示直线从左到右上升；负梯度表示下降。水平线的梯度为 0，垂直线的梯度未定义。
WJEC exam questions often ask you to find the gradient from a diagram or to determine whether a given triangle is right-angled by computing gradients of its sides.
WJEC 考试题目经常要求根据图表求梯度，或者通过计算三角形各边的梯度来判断它是否为直角三角形。
2. Distance and Midpoint Formulas | 距离与中点公式
The distance between two points A(x₁, y₁) and B(x₂, y₂) is found using Pythagoras' theorem in the coordinate plane:
在坐标平面中，两点 A(x₁, y₁) 和 B(x₂, y₂) 之间的距离可通过毕达哥拉斯定理求得：
AB = √[(x₂ − x₁)² + (y₂ − y₁)²]
The midpoint M of AB has coordinates that are the averages of the x‑ and y‑coordinates:
线段 AB 的中点 M 的坐标为 x 坐标和 y 坐标的平均值：
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
These formulas are essential for solving problems involving lengths of line segments, centres of circles, and properties of quadrilaterals. For example, to show that a quadrilateral is a parallelogram, you can prove that the diagonals bisect each other (their midpoints coincide).
这两个公式对于求解线段长度、圆心以及四边形性质的问题至关重要。例如，要证明一个四边形是平行四边形，可以证明其对角线互相平分（中点重合）。
In WJEC papers, you might be asked to find the distance between two points and then use it to calculate a perimeter or verify that a circle passes through a given point.
在 WJEC 试卷中，可能会要求你计算两点之间的距离，然后用它来计算周长或验证圆是否通过某个给定点。
3. Gradient and the Angle of Inclination | 梯度与倾斜角
The gradient m of a line is related to its angle of inclination θ measured from the positive x‑axis:
直线的梯度 m 与其相对于 x 轴正方向的倾斜角 θ 有关：
m = tan θ
This connection allows you to find the angle between two lines. If two lines have gradients m₁ and m₂, the acute angle α between them satisfies:
通过这个关系，你可以求得两条直线之间的夹角。如果两条直线的梯度分别为 m₁ 和 m₂，则它们之间的锐角 α 满足：
tan α = | (m₁ − m₂) / (1 + m₁ m₂) |
The denominator will be zero when the lines are perpendicular, which gives a separate condition (see next section). For WJEC, you only need the formula for the angle between two lines in its simplest form; questions typically involve finding the gradient that makes a specified angle with another line.
当分母为零时，两条直线互相垂直，此时有一个单独的条件（见下一节）。对于 WJEC，你只需要掌握两条直线夹角公式的最简单形式；题目通常涉及求与另一条直线形成指定角度的梯度。
4. Parallel and Perpendicular Lines | 平行线与垂直线
Two non‑vertical lines are parallel if and only if their gradients are equal:
两条非垂直线平行当且仅当它们的梯度相等：
m₁ = m₂
Two lines are perpendicular if and only if the product of their gradients is −1:
两条直线垂直当且仅当它们的梯度之积为 −1：
m₁ × m₂ = −1
These rules are used extensively to find equations of lines parallel or perpendicular to a given line, and to prove geometric results such as the orthocentre of a triangle. When a line is vertical, its gradient is undefined; a line perpendicular to it is horizontal (gradient 0).
这些规则被广泛用于求与给定直线平行或垂直的直线方程，以及证明几何结果（如三角形的垂心）。当一条直线垂直时，其梯度未定义；与之垂直的直线是水平的（梯度为 0）。
WJEC candidates should be comfortable working with gradients expressed as fractions and applying the negative reciprocal for perpendicular gradients quickly.
WJEC 考生应能熟练处理以分数表示的梯度，并迅速应用负倒数来求垂直梯度。
5. Equations of a Straight Line | 直线方程的各种形式
The equation of a straight line can be written in several useful forms. The most common is the gradient–intercept form:
直线方程有几种常用的形式。最常见的是斜截式：
y = mx + c
where m is the gradient and c is the y‑intercept. Another form is the point–gradient form, used when you know one point (x₁, y₁) on the line and its gradient:
其中 m 是梯度，c 是 y 轴截距。另一种形式是点斜式，用于已知直线上一点 (x₁, y₁) 和梯度的情况：
y − y₁ = m (x − x₁)
en these forms and identify the gradient from the general form: m = −a/b (provided b ≠ 0).
一般式是 ax + by + c = 0，这在 WJEC 的答案中经常要求使用。你必须能够在这些形式之间进行转换，并从一般式中识别梯度：m = −a/b（前提是 b ≠ 0）。
| Form | Equation | Key info |
|---|---|---|
| Gradient–intercept | y = mx + c | m, c directly visible |
| Point–gradient | y − y₁ = m(x − x₁) | Known point and gradient |
| Two–point | (y − y₁)/(x − x₁) = (y₂ − y₁)/(x₂ − x₁) | Two known points |
| General | ax + by + c = 0 | a, b, c integers (WJEC preference) |
ed to give your final line equation in the form ax + by + c = 0 with integer coefficients.
在 WJEC 考试中，经常会要求用一般式 ax + by + c = 0 给出最终的直线方程，且系数为整数。
6. Intersection of Lines | 直线的交点
To find the intersection point of two lines, solve their equations simultaneously. For non‑parallel lines, this yields a unique point (x, y).
要找到两条直线的交点，需同时求解它们的方程。对于非平行直线，可以得到一个唯一的点 (x, y)。
If the lines are given in general form:
如果直线以一般式给出：
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
you can use elimination or substitution. The point of intersection is the solution that satisfies both equations. In WJEC problems, you might need the intersection to find the circumcentre of a triangle (intersection of perpendicular bisectors) or to determine whether three lines are concurrent.
你可以使用消元法或代入法。交点是同时满足两个方程的解。在 WJEC 的题目中，你可能需要利用交点来求三角形的外心（垂直平分线的交点），或判断三条直线是否共点。
When two lines are parallel, the system has no solution. If the two equations represent the same line (multiples of each other), there are infinitely many solutions.
当两条直线平行时，方程组无解。如果两个方程代表同一条直线（互为倍数关系），则有无穷多解。
7. Circles: Equation and Centre–Radius Form | 圆：方程与圆心-半径式
The standard equation of a circle with centre (a, b) and radius r is:
以 (a, b) 为圆心、半径为 r 的圆的标准方程为：
(x − a)² + (y − b)² = r²
uation becomes x² + y² + 2gx + 2fy + c = 0, where the centre is (−g, −f) and the radius is √g² + f² − c. Completing the square is the key technique to convert from the expanded form to centre–radius form.
展开后，圆方程变为 x² + y² + 2gx + 2fy + c = 0，其中圆心为 (−g, −f)，半径为 √g² + f² − c。配方法是将展开形式转换为圆心-半径式的关键技巧。
WJEC exam items may ask you to show that a given equation represents a circle, find its centre and radius, or determine whether a point lies inside, on, or outside the circle by substituting coordinates into the left‑hand side and comparing with r².
WJEC 考题可能会要求你证明给定的方程表示一个圆，求出它的圆心和半径，或者通过将点的坐标代入方程左侧并与 r² 进行比较，来判断该点在圆内、圆上还是圆外。
8. Tangents and Chords of a Circle | 圆的切线与弦
The tangent to a circle at a given point is perpendicular to the radius at that point. To find the equation of the tangent, first determine the gradient of the radius, then use the negative reciprocal for the tangent's gradient, and apply the point–gradient form.
圆上某一点的切线垂直于该点处的半径。要求切线方程，首先确定半径的梯度，然后利用负倒数为切线的梯度，再使用点斜式。
If the circle's centre is C and the point of contact is P, the gradient of CP is mCP = (yP − yC) / (xP − xC). The gradient of the tangent is therefore mtan = −1 / mCP. For a tangent from an external point, you may need to use the fact that the distance from the centre to the tangent equals the radius.
如果圆心为 C，切点为 P，则 CP 的梯度为 mCP = (yP − yC) / (xP − xC)。因此切线的梯度为 mtan = −1 / mCP。对于从圆外一点引出的切线，你可能需要利用圆心到切线的距离等于半径这一性质。
The chord of a circle is a line segment connecting two points on the circumference. The perpendicular bisector of a chord always passes through the centre. WJEC often tests this property to find the centre or equation of a circle given a chord and a tangent condition.
圆的弦是连接圆周上两点的线段。弦的垂直平分线永远经过圆心。WJEC 经常利用这一性质，在给定弦和切线条件的情况下求圆心或圆的方程。
9. Intersection of a Line and a Circle | 直线与圆的交点
discriminant Δ = b² − 4ac of this quadratic tells you how many intersection points exist:
要找出直线与圆的交点，需联立求解它们的方程。将直线方程 y = mx + c（或 x = k）代入圆方程，得到一个关于一个变量的二次方程。该二次方程的判别式 Δ = b² − 4ac 可以说明交点的个数：
Δ > 0: two distinct intersection points (secant line)
Δ > 0：两个不同的交点（割线）
Δ = 0: the line is a tangent (touches the circle at exactly one point)
Δ = 0：直线为切线（恰好与圆相切于一点）
Δ < 0: no intersection (line misses the circle)
Δ < 0：无交点（直线与圆不相交）
tangent or to find the value of k for which the line y = mx + k is tangent to the circle.
这种判别式方法在 WJEC 的题目中特别有用，例如要求你证明某条直线是切线，或者求出使得直线 y = mx + k 与圆相切的 k 值。
10. Parametric Equations of Curves | 曲线的参数方程
Parametric equations express the coordinates x and y in terms of a third variable, usually t or θ. For a circle centred at the origin with radius r, the parametric equations are:
参数方程使用第三个变量（通常为 t 或 θ）来表示坐标 x 和 y。对于以原点为圆心、半径为 r 的圆，其参数方程为：
x = r cos θ, y = r sin θ
C includes parametric equations of curves such as parabolas and ellipses, and expects you to convert between parametric and Cartesian forms by eliminating the parameter.
圆心为 (a, b) 的圆的参数形式为 x = a + r cos θ, y = b + r sin θ。WJEC 包含抛物线、椭圆等曲线的参数方程，并要求你通过消去参数在参数形式和笛卡尔形式之间进行转换。
, to eliminate θ from the circle parametric equations, use the identity cos²θ + sin²θ = 1:
例如，要从圆的参数方程中消去 θ，可利用恒等式 cos²θ + sin²θ = 1：
(x − a)²/r² + (y − b)²/r² = 1 → (x − a)² + (y − b)² = r²
11. Loci in Coordinate Geometry | 坐标几何中的轨迹
A locus is the set of all points that satisfy a given condition. Common loci in WJEC exams include the perpendicular bisector of a segment (all points equidistant from two fixed points) and the angle bisector (all points equidistant from two intersecting lines).
轨迹是满足给定条件的所有点的集合。WJEC 考试中常见的轨迹包括线段的垂直平分线（到两个固定点距离相等的所有点）和角平分线（到两条相交直线距离相等的所有点）。
implifies to a linear equation. For two lines L₁ and L₂, the distance from a variable point (x, y) to each line is equated to find the angle bisectors.
对于点 A 和 B，垂直平分线由 PA² = PB² 给出，化简后得到一个线性方程。对于两条直线 L₁ 和 L₂，将动点 (x, y) 到每条直线的距离等式联立，即可求出角平分线。
Another typical WJEC locus problem is finding the path traced by the midpoint of a chord of a circle as the chord rotates, or the set of points from which tangents to a circle have equal length.
另一个典型的 WJEC 轨迹问题是，随着弦的旋转，求圆弦中点的轨迹；或者求使得到圆的两条切线长度相等的点的集合。
12. Exam Strategy and Common Pitfalls | 考试策略与常见错误
Start by sketching a diagram – it helps visualise geometric relationships and avoid sign errors. Label coordinates clearly and always state the formulas you are using; in WJEC, method marks are awarded for correct application even if arithmetic slips occur.
首先要画出示意图——这有助于将几何关系可视化，并避免符号错误。清晰标记坐标，并始终说明你使用的公式；在 WJEC 中，即使计算有误，正确应用方法也能获得方法分。
Be careful with signs when calculating gradients and distances, especially when coordinates are negative. When finding the equation of a tangent to a circle, many students forget that the radius is perpendicular to the tangent – draw the radius explicitly.
在计算梯度和距离时要注意符号，尤其是坐标为负时。在求圆的切线方程时，许多学生会忘记半径垂直于切线——请明确画出半径。
gers with no common factors (WJEC often specifies integer coefficients). Avoid leaving answers as decimals unless the question permits; exact fractional or surd forms are preferred.
用 ax + by + c = 0 的形式表示直线时，要确保 a、b 和 c 为整数且没有公因数（WJEC 经常要求整数系数）。除非题目允许，否则不要将答案保留为小数；最好使用精确的分数或根式形式。
Finally, always check that your answers make sense in the context of a diagram – does a tangent touch the circle? Is the midpoint coordinate realistic? This mental check can catch many careless mistakes.
最后，总是在图形背景下检查你的答案是否合理——切线是否与圆相切？中点坐标是否符合实际？这种心理检验可以捕捉到许多粗心造成的错误。
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