IB WJEC Mathematics: Coordinate Geometry Key Points | IB WJEC 数学：坐标几何 考点精讲

📚 IB WJEC Mathematics: Coordinate Geometry Key Points | IB WJEC 数学：坐标几何 考点精讲

Coordinate geometry, also known as analytic geometry, forms a cornerstone of the IB and WJEC mathematics syllabus. It bridges algebra and geometry, allowing us to describe lines, circles, and other curves through equations, and to analyse their properties with precision. Mastery of this topic is essential not only for examination success but also for deeper studies in calculus and vectors. This article consolidates the key concepts, common pitfalls, and examination techniques you need to excel.

坐标几何，又称解析几何，是 IB 与 WJEC 数学大纲的基石。它将代数与几何联系起来，使我们能够通过方程描述直线、圆和其他曲线，并精确分析它们的性质。熟练掌握这一主题不仅对考试成功至关重要，也为后续学习微积分和向量打下基础。本文梳理了核心概念、常见易错点以及你需要掌握的应试技巧。

1. The Distance Formula | 距离公式

The distance between two points A(x₁, y₁) and B(x₂, y₂) is derived from Pythagoras’ theorem and is given by AB = √[(x₂ − x₁)² + (y₂ − y₁)²]. This formula works in all four quadrants and remains the fundamental tool for finding lengths of line segments.

两点 A(x₁, y₁) 和 B(x₂, y₂) 之间的距离由勾股定理导出，公式为 AB = √[(x₂ − x₁)² + (y₂ − y₁)²]。该公式适用于所有四个象限，是计算线段长度的基本工具。

A common mistake is forgetting to square the differences or to take the square root at the end. Always check that the order of subtraction does not matter because the difference is squared, but consistency in labelling is helpful for avoiding sign errors.

常见错误是忘记将差值平方或最后忘记开平方根。请始终注意，由于差值会被平方，相减的顺序并不影响结果，但保持标示一致有助于避免符号错误。

2. The Midpoint Formula | 中点公式

The midpoint M of line segment AB has coordinates ((x₁ + x₂)/2, (y₁ + y₂)/2). This simple average of the x- and y-coordinates is invaluable for bisecting a segment or finding the centre of a shape.

线段 AB 的中点 M 坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。这个简单的 x 坐标与 y 坐标平均值在平分线段或求图形中心时非常有用。

When using the midpoint in circle or locus problems, remember that a diameter’s midpoint is the circle’s centre, linking this formula directly to the equation of a circle.

在圆或轨迹问题中使用中点时，请记住直径的中点就是圆心，这将该公式与圆的方程直接联系起来。

3. Gradient (Slope) of a Straight Line | 直线的斜率

The gradient m of the line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). Gradient measures steepness and direction; a positive gradient indicates an increasing line, a negative gradient a decreasing line, zero means horizontal, and undefined gradient (infinite) corresponds to a vertical line.

通过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率 m 公式为 m = (y₂ − y₁)/(x₂ − x₁)。斜率衡量倾斜程度和方向；正斜率表示上升直线，负斜率表示下降直线，零斜率表示水平线，斜率未定义（无穷大）对应于垂直线。

Do not divide by zero – a vertical line has equation x = constant and its gradient is undefined. Also, note that collinear points share the same gradient when taken in any pair.

不要除以零 —— 垂直线的方程是 x = 常数，其斜率无定义。此外，注意共线点任意两点连线的斜率都相等。

4. Equations of a Straight Line | 直线方程的各种形式

You must be fluent in three main forms: the gradient-intercept form y = mx + c, where m is the gradient and c is the y-intercept; the point-gradient form y − y₁ = m(x − x₁), used when you know a point and the slope; and the general form ax + by + d = 0. The WJEC specification often expects answers in a specific form, so read the question carefully.

你必须熟练掌握三种主要形式：斜截式 y = mx + c，其中 m 为斜率，c 为 y 轴截距；点斜式 y − y₁ = m(x − x₁)，在已知一点和斜率时使用；以及一般式 ax + by + d = 0。WJEC 考试大纲通常要求将答案写成特定形式，因此请仔细审题。

To find the equation of a line given two points, first calculate the gradient, then substitute one point into the point-gradient form. Avoid rounding slopes unless explicitly told to; use fractions to keep exact values.

给定两点求直线方程时，先计算斜率，再将其中一个点代入点斜式。除非明确要求，否则不要对斜率取近似值；使用分数以保持精确值。

5. Parallel and Perpendicular Lines | 平行线与垂直线

Two distinct lines are parallel if and only if their gradients are equal: m₁ = m₂. They are perpendicular if and only if the product of their gradients is −1: m₁ × m₂ = −1, provided neither line is vertical.

两条不同直线平行当且仅当它们的斜率相等：m₁ = m₂。两条直线垂直当且仅当它们的斜率之积为 −1：m₁ × m₂ = −1，前提是两条直线都不是垂直的。

Parallel: m₁ = m₂	平行：m₁ = m₂
Perpendicular: m₁ = −1/m₂	垂直：m₁ = −1/m₂

For vertical and horizontal lines: a vertical line x = a is parallel to any other vertical line and perpendicular to any horizontal line y = b.

对于垂直线和水平线：垂直线 x = a 与任何其他垂直线平行，并与任何水平线 y = b 垂直。

6. Intersection of Lines | 直线的交点

To find the point where two lines intersect, solve their equations simultaneously. The algebraic solution yields the coordinates of the common point. If the lines are parallel, there is no solution; if the equations represent the same line, there are infinitely many solutions.

要求两条直线的交点，需联立方程求解。代数解给出公共点的坐标。如果两线平行，则无解；如果方程表示同一条直线，则有无穷多解。

Substitution and elimination are both acceptable methods, but elimination often reduces arithmetic errors. Always check your intersection point by substituting it back into both original equations.

代入法和消元法均可接受，但消元法通常能减少算术错误。始终将求得的交点代回两个原方程进行检验。

7. Distance from a Point to a Line | 点到直线的距离

The perpendicular distance from a point P(x₁, y₁) to the line ax + by + c = 0 is given by d = |ax₁ + by₁ + c| / √(a² + b²). This formula is essential for calculating the shortest distance and appears often in circle geometry problems (e.g., distance from centre to a chord).

点 P(x₁, y₁) 到直线 ax + by + c = 0 的垂直距离公式为 d = |ax₁ + by₁ + c| / √(a² + b²)。该公式对于计算最短距离至关重要，常出现在圆几何问题中（例如圆心到弦的距离）。

Ensure the line equation is written with zero on one side before identifying a, b, and c. The absolute value guarantees a positive distance, and the denominator normalises the perpendicular component.

在确定 a、b 和 c 之前，请确保直线方程化为一边为零的形式。绝对值保证距离为正，分母将垂直分量归一化。

8. Equation of a Circle | 圆的方程

The standard form of a circle with centre (h, k) and radius r is (x − h)² + (y − k)² = r². Expanding gives the general form x² + y² + 2gx + 2fy + c = 0, where the centre is (−g, −f) and radius = √(g² + f² − c). To be a valid circle, g² + f² − c > 0.

圆心为 (h, k)、半径为 r 的圆的标准方程为 (x − h)² + (y − k)² = r²。展开得到一般式 x² + y² + 2gx + 2fy + c = 0，其中圆心为 (−g, −f)，半径 = √(g² + f² − c)。要使方程表示一个圆，须满足 g² + f² − c > 0。

Completing the square is the key skill to convert between forms. Watch out for signs: the centre (h, k) appears with opposites in (x − h) and (y − k); likewise in the general form, centre coordinates are −g and −f.

配方法是在两种形式间转换的关键技巧。注意符号：圆心 (h, k) 在 (x − h) 和 (y − k) 中以相反数出现；类似地，在一般式中，圆心坐标为 −g 和 −f。

9. Tangents and Chords of a Circle | 圆的切线与弦

A tangent touches the circle at exactly one point. The tangent at a point P on the circle is perpendicular to the radius at P. Therefore, find the gradient of the radius, then use the negative reciprocal to obtain the tangent gradient. The chord’s perpendicular bisector passes through the centre of the circle – a fact often used to find the centre from a chord’s endpoints.

切线仅与圆交于一点。圆上点 P 处的切线垂直于该点处的半径。因此，先求出半径的斜率，再取负倒数得到切线斜率。弦的垂直平分线经过圆心——这一性质常用于从弦的端点求圆心。

To find the length of a chord, draw the perpendicular from the centre to the chord; this bisects the chord. Then apply Pythagoras’ theorem using the radius and the perpendicular distance from the centre to the chord.

求弦长时，从圆心作弦的垂线，该垂线平分弦。再利用半径以及圆心到弦的垂直距离，应用勾股定理即可求解。

• If a line is a tangent, the perpendicular distance from the centre to the line equals the radius. This is a powerful condition in WJEC exam questions.

  若一条直线是切线，则圆心到该直线的垂直距离等于半径。这是 WJEC 考试题中一个强有力的条件。

• Discriminant method: substitute the line equation into the circle equation to form a quadratic; a tangent yields a discriminant Δ = 0.

  判别式法：将直线方程代入圆的方程得到一个二次方程；切线对应判别式 Δ = 0。

10. Parametric Equations and Loci | 参数方程与轨迹

Parametric equations express x and y separately in terms of a third variable, usually t or θ. For a circle centred at the origin, a common parametrisation is x = r cos θ, y = r sin θ. In WJEC and IB contexts, you may need to convert between parametric and Cartesian forms by eliminating the parameter.

参数方程用第三个变量（通常为 t 或 θ）分别表示 x 和 y。对于圆心在原点的圆，常见的参数形式为 x = r cos θ，y = r sin θ。在 WJEC 和 IB 考纲中，你可能需要通过消去参数在参数方程与直角坐标方程之间进行转化。

Locus problems describe the path of a point under given constraints. Translate the geometric condition into an algebraic equation relating coordinates. Common loci include perpendicular bisectors, angle bisectors, and circles defined by a fixed distance from a point.

轨迹问题描述一个点在给定约束下的路径。应将几何条件转化为关于坐标的代数方程。常见的轨迹包括垂直平分线、角平分线以及由一个定点定距定义的圆。

When eliminating a parameter, look for trigonometric identities such as sin²θ + cos²θ = 1, or solve for t from one equation and substitute into the other. Be careful with domain restrictions from the parameter range.

消参数时，利用三角恒等式如 sin²θ + cos²θ = 1，或从一个方程解出 t 代入另一个方程。注意由参数范围带来的定义域限制。

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