Equations of Straight Lines | 直线方程

📚 Equations of Straight Lines | 直线方程

Understanding the equations of straight lines is fundamental in A-Level Mathematics. It forms the basis for coordinate geometry, calculus applications, and modelling real-world relationships. This article covers all key concepts required for Edexcel A-Level, from gradient and various forms of the equation to parallel and perpendicular lines, distances, midpoints, intersections, and perpendicular bisectors.

理解直线方程是A-Level数学的基础，它为坐标几何、微积分应用以及现实世界关系的建模奠定了基础。本文涵盖了Edexcel A-Level要求的所有关键概念，从斜率和各种方程形式，到平行与垂直直线、距离、中点、交点以及垂直平分线。

1. Gradient of a Straight Line | 直线的斜率

The gradient (or slope) of a straight line measures its steepness and direction. For two points (x₁, y₁) and (x₂, y₂) on the line, the gradient m is calculated using the formula:

直线的斜率衡量其倾斜程度和方向。对于直线上的两点 (x₁, y₁) 和 (x₂, y₂)，斜率 m 使用以下公式计算：

m = (y₂ − y₁) / (x₂ − x₁)

A positive gradient means the line rises from left to right, a negative gradient means it falls, a zero gradient indicates a horizontal line, and an undefined gradient corresponds to a vertical line.

斜率为正表示直线从左到右上升，斜率为负表示下降，斜率为零表示水平线，斜率未定义对应于垂直线。

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2. Forms of the Equation: y = mx + c | 方程形式：y = mx + c

The most common form is the slope-intercept form: y = mx + c, where m is the gradient and c is the y-intercept (the point where the line crosses the y-axis, i.e. 0, c). This form is useful for quickly sketching a line and identifying its gradient and intercept.

最常见的形式是斜截式：y = mx + c，其中 m 是斜率，c 是 y 轴截距（即直线与 y 轴的交点，(0, c)）。这种形式便于快速绘制直线并识别其斜率和截距。

For example, the line y = 2x − 3 has a gradient of 2 and crosses the y-axis at (0, −3).

例如，直线 y = 2x − 3 的斜率为 2，与 y 轴交于点 (0, −3)。

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3. Point-Slope Form | 点斜式

When a line has a known gradient m and passes through a specific point (x₁, y₁), the equation can be written in point-slope form:

当直线已知斜率 m 且经过特定点 (x₁, y₁) 时，方程可以写成点斜式：

y − y₁ = m(x − x₁)

This form is extremely useful for finding the equation of a line when you are given a point and the gradient, or when working with tangents and normals in differentiation. You can then rearrange it into the y = mx + c form if needed.

此形式在给定点和斜率求直线方程时非常有用，也适用于微积分中的切线与法线问题。如有需要，可将其重新整理为 y = mx + c 的形式。

For instance, a line with gradient 4 passing through (1, 2) gives: y − 2 = 4(x − 1), which simplifies to y = 4x − 2.

例如，斜率为 4 并经过点 (1, 2) 的直线可写为：y − 2 = 4(x − 1)，整理得 y = 4x − 2。

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4. Finding the Equation from Two Points | 由两点求直线方程

To find the equation of a straight line passing through two points (x₁, y₁) and (x₂, y₂), first calculate the gradient m using the gradient formula, then substitute one point into the point-slope form, or directly into y = mx + c to find c.

要找到经过两点 (x₁, y₁) 和 (x₂, y₂) 的直线方程，首先使用斜率公式计算斜率 m，然后将其中一点代入点斜式，或直接代入 y = mx + c 求出 c。

Example: For points A(2, 5) and B(4, 9), m = (9 − 5) / (4 − 2) = 2. Using point A: y − 5 = 2(x − 2) → y = 2x + 1.

示例：对于点 A(2, 5) 和 B(4, 9)，m = (9 − 5) / (4 − 2) = 2。代入点 A：y − 5 = 2(x − 2) → y = 2x + 1。

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5. General Form and Intercepts | 一般式和截距

The general form of a straight line is written as ax + by + c = 0, where a, b and c are integers and a is usually positive. This form is handy for finding intersections and is often used in linear programming and vector geometry.

直线的一般式写作 ax + by + c = 0，其中 a、b 和 c 为整数，且 a 通常为正。此形式便于求交点，常用于线性规划和向量几何中。

To find the x-intercept, set y = 0 and solve for x; for the y-intercept, set x = 0 and solve for y. The intercepts can also be directly read from the intercept form x/p + y/q = 1, where p and q are the x- and y-intercepts respectively.

要求 x 轴截距，设 y = 0 并解出 x；要求 y 轴截距，设 x = 0 并解出 y。截距也可以直接从截距式 x/p + y/q = 1 中读出，其中 p 和 q 分别是 x 和 y 轴截距。

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6. Parallel Lines | 平行直线

Two distinct straight lines are parallel if and only if their gradients are equal. That is, m₁ = m₂. If the lines are given in general form, a₁/a₂ = b₁/b₂ ≠ c₁/c₂ indicates parallelism.

两条不同的直线平行的充分必要条件是它们的斜率相等，即 m₁ = m₂。如果直线以一般式给出，则 a₁/a₂ = b₁/b₂ ≠ c₁/c₂ 表示平行。

For example, y = 3x + 5 and y = 3x − 2 are parallel because both have gradient 3. The line parallel to 2x + 3y − 6 = 0 passing through (1, 1) has equation 2x + 3y = 5.

例如，y = 3x + 5 和 y = 3x − 2 是平行的，因为两者的斜率均为 3。与直线 2x + 3y − 6 = 0 平行且经过点 (1, 1) 的直线方程为 2x + 3y = 5。

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7. Perpendicular Lines | 垂直直线

Two lines are perpendicular if the product of their gradients is -1, provided both gradients are defined and non-zero. That is, m₁ × m₂ = −1, or equivalently m₂ = −1/m₁. Vertical and horizontal lines are also perpendicular (gradient undefined and zero).

如果两条直线的斜率之积为 -1，则它们垂直，前提是两者斜率均有定义且非零。即 m₁ × m₂ = −1，或等价地 m₂ = −1/m₁。垂直线与水平线也相互垂直（斜率未定义和零）。

This rule is essential for finding equations of normals to curves and for perpendicular bisectors.

这一法则对于求曲线的法线方程和垂直平分线至关重要。

Example: The line perpendicular to y = 2x + 3 passing through (4, 1) has gradient −½, giving equation y − 1 = −½(x − 4).

示例：垂直于 y = 2x + 3 且经过点 (4, 1) 的直线斜率为 −½，方程为 y − 1 = −½(x − 4)。

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8. Midpoint of a Line Segment | 线段的中点

The midpoint M of a line segment joining points (x₁, y₁) and (x₂, y₂) has coordinates given by the average of the x- and y-coordinates:

连接点 (x₁, y₁) 和 (x₂, y₂) 的线段的中点为 M，其坐标为 x 坐标和 y 坐标的平均值：

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

The midpoint formula is used extensively in coordinate geometry, particularly when working with perpendicular bisectors and centres of circles determined by endpoints of a diameter.

中点公式广泛用于坐标几何中，尤其是在处理垂直平分线以及由直径端点确定圆心时。

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9. Distance Between Two Points | 两点间的距离

The distance d between points (x₁, y₁) and (x₂, y₂) is derived from Pythagoras’ theorem:

两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离 d 由勾股定理导出：

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

This formula is vital for calculating the length of a line segment, proving that a triangle is right-angled (using the converse of Pythagoras), and for finding the radius of a circle given its centre and a point on the circumference.

该公式对于计算线段长度，证明三角形为直角三角形（利用勾股定理逆定理），以及已知圆心和圆周上一点求圆的半径至关重要。

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10. Intersection of Two Lines | 两条直线的交点

To find the point where two lines intersect, solve their equations simultaneously. If the equations are given in different forms, it is usually simplest to substitute the expression for y from one equation into the other, or to solve the linear system by elimination.

要找到两条直线的交点，需联立它们的方程。如果方程形式不同，通常最简单的做法是将一个方程中 y 的表达式代入另一个方程，或通过消元法解此线性方程组。

The intersection point is the unique solution unless the lines are parallel (no intersection) or coincident (infinitely many points). Checking the gradients first can tell you about the nature of the intersection.

除非直线平行（无交点）或重合（无穷多个交点），否则交点有唯一解。先检查斜率可以了解交点的性质。

For example, solving y = 2x + 1 and y = −x + 4 gives 2x + 1 = −x + 4 → 3x = 3 → x = 1, y = 3. Intersection at (1, 3).

例如，解 y = 2x + 1 和 y = −x + 4 得到 2x + 1 = −x + 4 → 3x = 3 → x = 1, y = 3。交点坐标为 (1, 3)。

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11. Perpendicular Bisectors | 垂直平分线

The perpendicular bisector of a line segment AB is a line that passes through the midpoint of AB and is perpendicular to AB. To find its equation: calculate the midpoint M of AB, find the gradient of AB, then take the negative reciprocal to get the gradient of the bisector, and finally use point-slope form with M.

线段 AB 的垂直平分线是一条经过 AB 中点且垂直于 AB 的直线。求其方程的步骤为：计算 AB 的中点 M，求出 AB 的斜率，然后取负倒数得到平分线的斜率，最后利用点斜式结合 M 写出方程。

This concept is crucial in locating the centre of a circle passing through two or three points, and in constructing Voronoi diagrams.

这一概念对于确定经过两点或三点的圆的圆心，以及构造 Voronoi 图至关重要。

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12. Modelling with Straight Lines | 直线模型应用

Straight line equations are frequently used to model linear relationships between variables in real-world contexts, such as converting between temperature scales (Celsius and Fahrenheit), analysing cost functions in business, or predicting values from a linear trend in data. The gradient represents the rate of change, and the intercept represents the starting value.

直线方程常被用来模拟现实世界中变量之间的线性关系，例如温度标度（摄氏和华氏）的转换、分析商业中的成本函数，或根据数据的线性趋势进行预测。斜率代表变化率，截距代表起始值。

In such models, you are often required to interpret the gradient and intercept in context, and sometimes to solve for one variable given the other. Always ensure the values obtained are sensible within the practical constraints of the problem.

在这类模型中，通常要求根据情境解释斜率和截距的含义，有时也需要已知一个变量求另一个变量。务必确保所得数值在问题的实际约束范围内是合理的。

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