📚 Coordinate Geometry: Key Points for IB/AQA Maths | 坐标几何考点精讲
Coordinate geometry unites algebra and geometry, allowing you to describe lines, curves, and shapes using equations. In IB and AQA Maths, mastering the distance formula, midpoint, gradient, equations of straight lines, perpendicular/parallel conditions, and circle geometry is essential. This guide summarises these core concepts with examples and exam tips.
坐标几何结合了代数与几何，让学生能够用方程描述直线、曲线和图形。在IB 和 AQA 数学考试中，掌握距离公式、中点、斜率、直线方程的多种形式、平行与垂直条件以及圆的几何性质是核心要点。本讲带你逐一梳理解题关键。
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.
两点 A(x1, y1) 与 B(x2, y2) 之间的距离由勾股定理导出。
d = √[(x2 − x1)2 + (y2 − y1)2]
For example, if A(3, 4) and B(7, 1), then d = √[(7−3)2 + (1−4)2] = √16 + 9 = √25 = 5.
例如，A(3, 4) 和 B(7, 1) 之间的距离 d = √[(7−3)2 + (1−4)2] = √16+9 = √25 = 5。
Always take the positive square root. This formula works in all quadrants.
永远取正值平方根。该公式适用于所有象限。
2. Midpoint Formula | 中点公式
The midpoint M of the segment joining (x1, y1) and (x2, y2) is found by averaging the coordinates.
线段两端点 (x1, y1) 和 (x2, y2) 的中点 M 由坐标平均值得到。
M = ((x1 + x2)/2, (y1 + y2)/2)
For points (2, −3) and (6, 5), the midpoint is ((2+6)/2, (−3+5)/2) = (4, 1).
对于 (2, −3) 和 (6, 5)，中点为 ((2+6)/2, (−3+5)/2) = (4, 1)。
This is useful for finding the centre of a line segment or verifying parallelogram properties.
这在求线段中心或验证平行四边形性质时很有用。
3. Gradient (Slope) of a Line | 直线的斜率
The gradient m measures steepness and direction.
斜率 m 度量直线的倾斜程度和方向。
m = (y2 − y1) / (x2 − x1)
horizontal line has m = 0, and a vertical line has undefined gradient (division by zero).
正斜率表示从左到右上升；负斜率表示下降。水平线斜率为 0，垂直线斜率无定义（除数为零）。
8), m = (8−2)/(4−1) = 6/3 = 2. For A(−2, 5) and B(3, 2), m = (2−5)/(3−(−2)) = −3/5 = −0.6.
已知 A(1,2) 和 B(4,8)，m = (8−2)/(4−1) = 2；若 A(−2,5) 和 B(3,2)，则 m = (2−5)/(3−(−2)) = −3/5。
4. Equations of a Straight Line | 直线方程
There are three common forms – gradient-intercept, point-gradient, and general form.
直线方程有三种常用形式：斜截式、点斜式和一般式。
y = mx + c (gradient−intercept form)
m is the gradient, c is the y-intercept.
m 为斜率，c 为 y 轴截距。
y − y1 = m(x − x1) (point−gradient form)
Use this when you know one point (x1, y1) and the gradient m.
当你已知一点 (x1, y1) 和斜率 m 时使用。
Ax + By + C = 0 (general form)
A, B, C are integers, often with A positive.
A、B、C 为整数，通常使 A 为正数。
Example: A line passes through (2, 3) with gradient 4 → y − 3 = 4(x − 2) → y = 4x − 5.
例：过点 (2,3) 且斜率为 4 → y − 3 = 4(x − 2) → y = 4x − 5。
5. Parallel and Perpendicular Lines | 平行线与垂直线
Two lines are parallel if and only if their gradients are equal: m1 = m2.
两直线平行当且仅当它们的斜率相等：m1 = m2。
Two lines are perpendicular if the product of their gradients is −1: m1 × m2 = −1. (Also, a vertical line is perpendicular to a horizontal line.)
两直线垂直当斜率乘积为 −1：m1 × m2 = −1。（竖直线与水平线也互相垂直。）
Example: Line L has gradient 2. A line parallel to L also has gradient 2. A line perpendicular to L has gradient −1/2.
例：直线 L 的斜率为 2，与它平行的直线斜率也是 2；与它垂直的直线斜率为 −1/2。
is y − 2 = 3(x − 1) → y = 3x − 1. Its perpendicular through (1,2) is y − 2 = −1/3 (x − 1).
已知直线 y = 3x + 7，过 (1,2) 的平行线为 y − 2 = 3(x − 1) → y = 3x − 1；过 (1,2) 的垂线为 y − 2 = −1/3 (x − 1)。
6. Distance from a Point to a Line | 点到直线的距离
For a line in general form Ax + By + C = 0 and a point P(x1, y1), the shortest distance is:
对于一般式直线 Ax + By + C = 0 和点 P(x1, y1)，最短距离为：
d = |Ax1 + By1 + C| / √A2 + B2
Example: Find the distance from (2, 3) to 3x − 4y + 5 = 0. A=3, B=−4, C=5. d = |3·2 + (−4)·3 + 5| / √9 + 16 = |6 − 12 + 5|/5 = |−1|/5 = 0.2.
例：求点 (2,3) 到直线 3x − 4y + 5 = 0 的距离。A=3, B=−4, C=5，d = |3×2 + (−4)×3 + 5| / √9+16 = |6−12+5|/5 = 1/5 = 0.2。
This formula is very useful for questions involving circles and tangents.
此公式在涉及圆与切线的题目中非常实用。
7. Equation of a Circle | 圆的方程
A circle with centre (h, k) and radius r has the standard form:
以 (h, k) 为圆心、半径为 r 的圆的标准方程为：
(x − h)2 + (y − k)2 = r2
The general form is x2 + y2 + 2gx + 2fy + c = 0, where centre is (−g, −f) and radius = √g2 + f2 − c, provided g2 + f2 − c > 0.
一般式为 x2 + y2 + 2gx + 2fy + c = 0，圆心为 (−g, −f)，半径 r = √g2 + f2 − c，且要求 g2 + f2 − c > 0。
Complete the square to convert between forms. For instance, x2 + y2 − 6x + 8y + 9 = 0 → (x−3)2 + (y+4)2 = 16 → centre (3,−4), r=4.
通过配方法转换形式。例如 x2 + y2 − 6x + 8y + 9 = 0 → (x−3)2 + (y+4)2 = 16，圆心 (3,−4)，半径 4。
8. Intersections of Lines and Circles | 直线与圆的交点
To find the intersection points, substitute the linear equation into the circle's equation and solve the resulting quadratic.
求交点时，将直线方程代入圆的方程，解所得二次方程。
The discriminant Δ = b2 − 4ac determines the number of intersection points:
判别式 Δ = b2 − 4ac 决定交点数：
If Δ > 0: two distinct intersection points (line is a secant).
若 Δ > 0：有两个不同的交点（直线为割线）。
If Δ = 0: exactly one intersection point (line is a tangent).
若 Δ = 0：恰有一个交点（直线为切线）。
If Δ < 0: no intersection (line does not meet the circle).
若 Δ < 0：无交点。
Example: Find intersections of x + y = 5 and x2 + y2 = 25. Substitute y = 5 − x → x2 + (5−x)2 = 25 → 2x2 −10x + 25 = 25 → 2x2 −10x = 0 → 2x(x−5)=0 → x=0 or x=5. Points: (0,5) and (5,0).
例：求 x + y = 5 与 x2 + y2 = 25 的交点。代入 y = 5 − x → x2 + (5−x)2 = 25 → 2x2 −10x = 0 → 交点 (0,5) 和 (5,0)。
9. Tangent to a Circle | 圆的切线
At the point of tangency, the radius is perpendicular to the tangent. This gives two main methods to find the tangent equation:
在切点处，半径与切线垂直。由此可使用以下两种主要方法求切线方程：
Method 1 – Using perpendicular gradient: Find the gradient of the radius OP (centre O to point P on the circle), then the tangent gradient mtan = −1/mOP, and use point-gradient form.
方法一：利用垂直斜率。求出半径 OP（圆心 O 到圆上点 P）的斜率，切线斜率 mtan = −1/mOP，再使用点斜式求方程。
Method 2 – le equation and set discriminant Δ = 0 to find c (if m is known) or m (if point is known).
方法二：判别式法。将直线方程 y = mx + c 代入圆的方程，令判别式 Δ = 0，解出 c（已知斜率时）或 m（已知点时）。
Example: Find tangent to x2 + y2 = 4/3. Tangent gradient = −3/4. Equation: y − 4 = −3/4(x − 3), simplifies to 3x + 4y = 25.
例：求圆 x2 + y2 = 25 在点 (3,4) 的切线。圆心 (0,0)，半径斜率 4/3，切线斜率 −3/4，方程 y − 4 = −3/4(x − 3) → 3x + 4y = 25。
10. Exam Tips & Common Mistakes | 考试技巧与常见错误
Checklist for coordinate geometry problems:
坐标几何解题注意事项：
Always ensure the line is in the correct form before reading off gradient or intercept. Rearrange if necessary.
读取斜率或截距前务必确认直线是否为正确形式，必要时要重新整理。
When using the distance formula, remember to square the differences inside the root – do not forget the brackets.
使用距离公式时，记住括号内差值的平方，不要遗漏括号导致符号错误。
For perpendicular lines, check that m1 × m2 = −1 exactly, not just 'negative reciprocal'. Be careful with fractions.
垂直条件必须严格满足 m1 × m2 = −1，不仅仅是'负倒数'，注意分数运算。
When completing the square for circle equations, add the same constants to both sides to avoid changing the equation.
在圆的方程中配方时，等式两边同时加上相同的常数项以保持恒等。
Use the discriminant to prove tangency without finding the point of contact – a common time‑saver in exams.
可使用判别式证明相切而无需找出切点——这是考试中常用的省时方法。
Double‑check the sign of C in the line equation when applying the point‑line distance formula; many students lose marks by forgetting the absolute value.
应用点到直线距离公式时，注意 C 的符号及绝对值，很多考生因忘加绝对值而丢分。
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