Coordinate Geometry | 坐标几何考点精讲

📚 Coordinate Geometry | 坐标几何考点精讲

Coordinate geometry, also known as analytic geometry, is a fundamental topic in OCR A-Level Mathematics. It bridges algebra and geometry, allowing you to describe geometric figures using equations and to solve geometric problems using algebraic techniques. Mastery of coordinate geometry is essential not only for the pure mathematics component but also for mechanics and further study. This revision guide covers all key concepts, from the basics of points, lines, and circles to advanced applications involving tangents, intersections, and loci, tailored to the OCR specification.

坐标几何，又称解析几何，是OCR A-Level数学中的核心主题。它连接了代数与几何，使你能够用方程描述几何图形，并运用代数方法解决几何问题。掌握坐标几何不仅对纯数学部分至关重要，对力学和后续学习也同样关键。这份复习指南涵盖了从点、直线和圆的基础到切线、交点和轨迹等高级应用的所有核心概念，紧密贴合OCR考纲。

1. Points, Distance, Midpoint, and Slope | 点、距离、中点与斜率

The most basic elements in coordinate geometry are points in the Cartesian plane, typically denoted as (x₁, y₁) and (x₂, y₂). Three essential calculations you must know are the distance between two points, the midpoint of a segment, and the slope (gradient) of the line joining them.

坐标几何中最基本的元素是笛卡尔平面上的点，通常记为(x₁, y₁)和(x₂, y₂)。你必须掌握的三个基本计算是两点间距离、线段中点以及连接两点的直线斜率（梯度）。

The distance d between (x₁, y₁) and (x₂, y₂) is given by the formula derived from Pythagoras’ theorem:

两点(x₁, y₁)与(x₂, y₂)之间的距离d由勾股定理推导出的公式给出：

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

This distance is always positive and represents the length of the straight-line segment joining the two points. When applying it, ensure you subtract coordinates consistently and square the differences before adding.

该距离始终为正，表示连接两点的直线段长度。应用时，请确保一致地相减坐标，在相加前将差值平方。

The midpoint M of the segment joining (x₁, y₁) and (x₂, y₂) is simply the average of the coordinates:

连接(x₁, y₁)和(x₂, y₂)的线段的中点M就是坐标的平均值：

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

This is especially useful for finding the centre of a circle given the endpoints of a diameter, or for checking symmetry in a diagram.

这在已知直径端点求圆心，或检验图形对称性时格外有用。

The slope (gradient) m of the line passing through (x₁, y₁) and (x₂, y₂) measures its steepness:

通过(x₁, y₁)和(x₂, y₂)的直线的斜率（梯度）m衡量其倾斜程度：

m = (y₂ − y₁) / (x₂ − x₁)

A positive gradient means the line rises as x increases; a negative gradient means it falls. A horizontal line has m = 0, and a vertical line has an undefined gradient (denominator zero).

正梯度表示直线随x增大而上升；负梯度表示下降。水平线的m = 0，铅垂线的梯度无定义（分母为零）。

2. Equation of a Straight Line | 直线方程

The equation of a straight line can be expressed in several forms, each useful in different contexts. The most common is the gradient-intercept form:

直线方程可以用多种形式表示，每种在不同情境下各有用途。最常见的是斜截式：

y = mx + c

Here m is the gradient and c is the y-intercept (where the line crosses the y-axis). This form is ideal for quickly sketching a line or identifying its gradient and y-intercept directly from the equation.

其中m为梯度，c为y轴截距（直线与y轴的交点）。这种形式非常适合快速画图或直接从方程识别梯度和截距。

Another important form is the point-gradient form, used when you know one point (x₁, y₁) on the line and the gradient m:

另一种重要形式是点斜式，当你知道直线上一点(x₁, y₁)和梯度m时使用：

y − y₁ = m(x − x₁)

This is particularly handy for finding the equation of a line parallel or perpendicular to a given line passing through a specific point. Simply substitute the known point and the appropriate gradient.

这在求通过某点且与给定直线平行或垂直的直线方程时特别方便。只需代入已知点和相应梯度即可。

The general form ax + by + c = 0 is also frequently used, especially when dealing with intersection points and systems of equations. To find the gradient from this form, rearrange to y = −(a/b)x − (c/b), so m = −a/b.

一般式ax + by + c = 0也常用，尤其是在处理交点和方程组时。要从此形式求梯度，可变形为y = −(a/b)x − (c/b)，因此m = −a/b。

3. Parallel and Perpendicular Lines | 平行与垂直线

Parallel lines never meet and have the same gradient. If line L₁ has gradient m₁ and line L₂ has gradient m₂, then L₁ ∥ L₂ ⇔ m₁ = m₂. Conversely, if two lines are parallel, their gradients are equal (provided they are not vertical).

平行线永不相交，且梯度相同。若直线L₁梯度为m₁，直线L₂梯度为m₂，则L₁ ∥ L₂ ⇔ m₁ = m₂。反之，若两直线平行，它们的梯度相等（假设不垂直）。

Perpendicular lines intersect at right angles. The condition for perpendicularity is that the product of their gradients equals −1: m₁ × m₂ = −1, or equivalently m₂ = −1/m₁. Vertical and horizontal lines are a special case: a vertical line (undefined gradient) is perpendicular to a horizontal line (gradient 0).

垂直线以直角相交。垂直的条件是它们梯度的乘积等于−1：m₁ × m₂ = −1，或等价地m₂ = −1/m₁。铅垂线与水平线是特例：铅垂线（梯度无定义）与水平线（梯度0）垂直。

When solving problems, always check whether lines are parallel or perpendicular to simplify calculations. For instance, the normal to a curve at a point is perpendicular to the tangent at that point.

解题时，始终检查直线是否平行或垂直以简化计算。例如，曲线上某点的法线垂直于该点处的切线。

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

The perpendicular distance from a point P(x₁, y₁) to a line L: ax + by + c = 0 is given by the formula:

点P(x₁, y₁)到直线L: ax + by + c = 0的垂直距离由以下公式给出：

d = |ax₁ + by₁ + c| / √(a² + b²)

This formula is essential for finding the shortest distance from a point to a line, which can be applied in topics like circle geometry (distance from centre to chord bisector) and optimisation problems.

该公式对于求点到直线的最短距离至关重要，可应用于圆的几何（圆心到弦的中垂线距离）和优化问题中。

To use it, ensure the line equation is in general form with zero on one side. Also remember the absolute value in the numerator guarantees a positive distance. The denominator is the magnitude of the normal vector of the line.

使用时，确保直线方程为一般形式，即一边为零。同样记住分子中的绝对值保证距离为正。分母是直线法向量的模。

5. Intersection of Lines | 直线的交点

To find the point of intersection of two lines, solve their equations simultaneously. If the lines are given in the form y = m₁x + c₁ and y = m₂x + c₂, set them equal to solve for x, then substitute back for y. For lines in general form ax + by + c = 0, use elimination or substitution methods.

要求两条直线的交点，联立求解其方程。若直线以y = m₁x + c₁和y = m₂x + c₂的形式给出，令它们相等解x，然后回代求y。对于一般形式ax + by + c = 0的直线，使用消元法或代入法。

If the lines are parallel (m₁ = m₂), there is either no solution (distinct parallel lines) or infinitely many solutions (the same line). The intersection of a pair of lines can also be found using matrix methods or Cramer’s rule, if you prefer.

若直线平行（m₁ = m₂），则可能无解（不同平行线）或无穷多解（同一直线）。也可以使用矩阵方法或克莱姆法则求直线对的交点。

Finding the intersections of lines is a building block for more complex work, such as determining the vertices of a triangle given its sides, or locating the intersection of a tangent and a circle’s radius.

求直线的交点是更复杂工作的基础，例如已知三角形三边求顶点，或定位切线与圆半径的交点。

6. Equation of a Circle | 圆的方程

A circle is defined as the set of all points at a fixed distance r (radius) from a fixed centre C(a, b). The standard form of the circle’s equation is:

圆定义为到固定中心C(a, b)距离等于定长r（半径）的所有点的集合。圆的标准方程为：

(x − a)² + (y − b)² = r²

From this, you can identify the centre (a, b) and radius r immediately. The squared terms have coefficients 1, and there is no xy term.

由此可立即识别圆心(a, b)和半径r。平方项系数为1，且不含xy项。

Sometimes the equation is given in expanded general form:

有时方程以展开的一般式给出：

x² + y² + 2gx + 2fy + c = 0

In this form, the centre is (−g, −f), and the radius is √(g² + f² − c). For a valid circle, the expression g² + f² − c must be positive. If it equals zero, the circle degenerates to a point; if negative, no real circle exists.

在此形式下，圆心为(−g, −f)，半径为√(g² + f² − c)。要得到有效圆，表达式g² + f² − c必须为正。若等于零，圆退化为一个点；若为负，不存在实圆。

Completing the square is the standard technique to convert from general to standard form. Always check that the coefficients of x² and y² are equal (usually 1) before starting.

配方法是将一般式转换为标准式的标准技巧。开始前务必检查x²和y²的系数是否相等（通常为1）。

7. Intersection of a Line and a Circle | 直线与圆的交点

To find where a line cuts a circle, substitute the line’s equation (e.g., y = mx + c) into the circle’s equation. This yields a quadratic equation in x (or y). The discriminant Δ = b² − 4ac of this quadratic tells you the nature of the intersection:

要求直线与圆的交点，将直线方程（如y = mx + c）代入圆的方程。这会得到一个关于x（或y）的一元二次方程。该二次方程的判别式Δ = b² − 4ac告诉你交点的性质：

Δ > 0 → two distinct points of intersection (the line is a secant).
Δ > 0 → 两个相异的交点（直线为割线）。
Δ = 0 → one point of intersection (the line is a tangent).
Δ = 0 → 一个交点（直线为切线）。
Δ < 0 → no intersection (the line misses the circle).
Δ < 0 → 无交点（直线不与圆相交）。

The discriminant method is powerful because it does not require solving for the actual points, only determining the relationship between the line and the circle. For finding exact intersection coordinates, solve the quadratic.

判别式法功能强大，因为它无需实际求出交点坐标，仅需判断直线与圆的关系。要找到具体交点坐标，须求解二次方程。

Remember that if the line goes through the centre, the chord becomes a diameter, and the midpoint of the chord is the centre itself. The perpendicular from the centre to a chord bisects the chord.

记住如果直线经过圆心，弦就变成直径，且弦的中点即为圆心本身。从圆心到弦的垂线平分该弦。

8. Tangents to a Circle | 圆的切线

A tangent to a circle is a line that touches the circle at exactly one point, called the point of tangency. The tangent is always perpendicular to the radius drawn to the point of contact. Therefore, if you know the centre C and the point of contact P, the gradient of the radius CP is m_r, and the gradient of the tangent is m_t = −1/m_r (provided m_r is defined).

圆的切线是一条恰好接触圆于一点（称为切点）的直线。切线总是垂直于连接切点的半径。因此，若已知圆心C及切点P，半径CP的梯度为m_r，则切线的梯度为m_t = −1/m_r（当m_r有定义时）。

To find the equation of a tangent at a given point on the circle, use the point-gradient form with the perpendicular gradient. For example, if the circle is (x − a)² + (y − b)² = r² and the point of tangency is T(x₁, y₁), the tangent line is:

要求圆上已知点处的切线方程，使用垂直梯度的点斜式即可。例如，若圆的方程为(x − a)² + (y − b)² = r²且切点为T(x₁, y₁)，则切线为：

(x₁ − a)(x − x₁) + (y₁ − b)(y − y₁) = 0

This can be derived using the fact that the radius and tangent are perpendicular, or via implicit differentiation if you have studied it.

这可通过半径与切线垂直的性质推得，若已学过隐函数求导也可由此导出。

When finding tangents from an external point Q(x₀, y₀) to a circle, there are generally two tangents. The process involves setting up the line equation through Q with unknown gradient m, using the condition that the perpendicular distance from the centre to the line equals the radius, and solving for m. Alternatively, use the discriminant method after substituting the line equation into the circle’s equation and setting Δ = 0.

当从外部点Q(x₀, y₀)引圆的切线时，通常有两条切线。方法是建立通过Q且梯度m未知的直线方程，利用圆心到该直线的垂直距离等于半径的条件求出m。另一种方法是代入直线方程到圆的方程后令Δ = 0求解。

9. Parametric Equations and Coordinate Geometry | 参数方程与坐标几何

Parametric equations allow you to describe curves by expressing x and y in terms of a third variable, typically t or θ. For a circle with centre (0,0) and radius r, a common parametric representation is:

参数方程通过用第三个变量（常为t或θ）表示x和y来描述曲线。对于圆心在(0,0)、半径为r的圆，常见的参数表示为：

x = r cos θ, y = r sin θ, 0 ≤ θ < 2π

If the centre is (a, b), the equations become x = a + r cos θ, y = b + r sin θ. This form is useful for modelling circular motion and for integration problems.

若圆心在(a, b)，参数方程变为x = a + r cos θ, y = b + r sin θ。这种形式在模拟圆周运动和积分问题中很有用。

Lines can also be expressed parametrically. Given a point (x₀, y₀) on the line and a direction vector (p, q), the parametric form is:

直线也可以用参数方程表示。给定直线上一点(x₀, y₀)及方向向量(p, q)，参数形式为：

x = x₀ + pt, y = y₀ + qt, t ∈ R

Eliminating the parameter t gives the Cartesian equation. The gradient of this line is q/p (provided p ≠ 0).

消去参数t即得笛卡尔方程。该直线的梯度为q/p（当p ≠ 0时）。

Parametric representations often simplify finding intersections: substitute the parametric coordinates into the other curve’s equation and solve for the parameter.

参数表示常能简化求交点的过程：将参数坐标代入另一曲线的方程并解出参数即可。

10. Locus Problems | 轨迹问题

A locus is a set of points that satisfy a given condition. In coordinate geometry, you often need to find the equation of a locus. Common loci include the perpendicular bisector of a line segment, the angle bisector of two lines, and circles defined by a fixed distance from a point.

轨迹是满足给定条件的点的集合。在坐标几何中，常常需要求出轨迹的方程。常见的轨迹包括线段的垂直平分线、两直线的角平分线，以及到定点距离为常数的圆。

For example, the locus of points equidistant from two fixed points A and B is the perpendicular bisector of segment AB. To find its equation, express PA² = PB² using coordinates, simplify, and obtain the equation of a straight line.

例如，到两定点A、B距离相等的点的轨迹是线段AB的垂直平分线。要求其方程，利用坐标表示PA² = PB²，化简后得到直线方程。

Another classic problem: the locus of points P such that the angle APB is a right angle (where A and B are fixed) is a circle with AB as diameter (excluding A and B themselves). This follows from the fact that the angle in a semicircle is a right angle.

另一个经典问题：使得∠APB为直角的点P的轨迹（A、B为定点）是以AB为直径的圆（不包括A、B点本身）。这源自半圆上的圆周角为直角的定理。

When approaching loci problems, introduce coordinates for the variable point P(x, y), translate the geometric condition into an equation involving x and y, and simplify. The resulting equation describes the locus.

解决轨迹问题时，为动点P(x, y)引入坐标，将几何条件转化为关于x和y的方程，并化简。所得方程即描述该轨迹。

11. Problem-Solving and Exam Tips | 解题与应试技巧

Coordinate geometry questions in OCR exams often combine several concepts in one problem. A typical question might give you the equation of a circle and a line with an unknown parameter, and ask you to find the value(s) that make the line a tangent, or to compute the coordinates of intersection and the length of a chord.

OCR考试中的坐标几何题目常在一问中综合多个概念。典型题目可能会给出一个圆和含未知参数的直线，要求求出使该直线成为切线的参数值，或计算交点坐标及弦长。

Always draw a clear diagram, even a rough sketch, to visualise the given information. This helps you understand relationships and avoid sign errors. Label the centre, points, and relevant radii or gradients.

始终画出清晰的示意图，哪怕是简图，以可视化已知信息。这有助于理解关系并避免符号错误。标出圆心、点以及相关半径或梯度。

Remember the important geometric properties: the radius to the point of tangency is perpendicular to the tangent; the perpendicular from the centre to a chord bisects the chord; the midpoint of the diameter is the centre. These facts often provide shortcuts in solutions.

牢记重要的几何性质：切点处的半径垂直于切线；圆心到弦的垂线平分弦；直径的中点为圆心。这些事实常能为解题提供捷径。

When using the discriminant, substitute correctly and expand carefully—algebraic slip-ups are common. Double-check that the equation is set to zero before identifying a, b, and c for the quadratic formula. For tangent from an external point, using the distance formula (centre to line = radius) can be more efficient than the discriminant.

使用判别式时，正确代入并仔细展开——代数失误很常见。在确认二次方程的a、b、c之前，再次检查方程是否已设为零。对于从外部点引切线，使用距离公式（圆心到直线的距离 = 半径）可能比判别式更高效。

Practise with past papers and time yourself. Recognising standard patterns, like converting between general and standard form of a circle, or finding the perpendicular bisector, will save valuable time in the exam.

用往年真题练习并计时。识别标准模式，如在圆的一般式和标准式之间转换，或求垂直平分线，将在考试中节省宝贵时间。

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Coordinate Geometry | 坐标几何考点精讲