Coordinate Geometry in the Plane | 平面坐标系中的图形与计算

📚 Coordinate Geometry in the Plane | 平面坐标系中的图形与计算

Coordinate geometry, also known as analytic geometry, connects algebra and geometry by describing shapes through equations. In this article, we explore key formulas and techniques for solving problems involving lines, circles, and areas in the Cartesian plane.

平面坐标系中的解析几何将代数与几何联系起来，用方程描述图形。本文将探讨笛卡尔平面中直线、圆以及面积等问题的核心公式与解题技巧。

1. Cartesian Coordinates and Distance | 直角坐标与距离

The Cartesian plane is defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). A point P is written as (x, y), where x is the horizontal displacement from the origin and y is the vertical displacement.

直角平面由两条互相垂直的数轴组成：x轴（水平）和y轴（垂直）。点P记为(x, y)，其中x是从原点出发的水平位移，y是垂直位移。

Given two points A(x₁, y₁) and B(x₂, y₂), the distance between them is derived from Pythagoras’ theorem:

已知两点A(x₁, y₁)和B(x₂, y₂)，它们之间的距离由勾股定理得出：

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

For example, the distance between (1, 2) and (4, 6) is √[(4−1)² + (6−2)²] = √[9 + 16] = 5.

例如，点(1, 2)与(4, 6)之间的距离为√[(4−1)² + (6−2)²] = √[9 + 16] = 5。

2. Midpoint Formula | 中点公式

The midpoint M of a segment connecting A(x₁, y₁) and B(x₂, y₂) has coordinates equal to the average of the endpoints’ coordinates:

连接A(x₁, y₁)和B(x₂, y₂)的线段中点M的坐标等于两端点坐标的平均值：

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

This formula is essential for finding centres of circles, medians of triangles, and points of symmetry.

该公式在求圆心、三角形中线以及对称点时非常关键。

3. Gradient (Slope) of a Line | 直线的斜率

The gradient measures the steepness of a line. For two points A(x₁, y₁) and B(x₂, y₂), the gradient m is:

斜率衡量一条直线的倾斜程度。对于两点A(x₁, y₁)和B(x₂, y₂)，斜率m为：

m = (y₂ − y₁) / (x₂ − x₁)

A positive gradient means the line rises from left to right; a negative gradient means it falls. A horizontal line has m = 0, while a vertical line has an undefined gradient.

斜率为正表示直线从左到右上升；斜率为负表示直线下降。水平线的斜率为0，而垂直线的斜率不存在（无定义）。

4. Equations of a Straight Line | 直线方程

There are several common forms for a linear equation. The point-slope form is useful when you know a point (x₁, y₁) and the gradient m:

直线方程有几种常见形式。若已知一点(x₁, y₁)和斜率m，可使用点斜式：

y − y₁ = m(x − x₁)

The slope-intercept form is y = mx + c, where c is the y-intercept. The general form is ax + by + d = 0.

斜截式为y = mx + c，其中c是y轴截距。一般式为ax + by + d = 0。

For example, the line through (2, 3) with gradient −1 has equation y − 3 = −1(x − 2), which simplifies to y = −x + 5.

例如，过点(2, 3)且斜率为−1的直线方程为y − 3 = −1(x − 2)，化简得y = −x + 5。

5. Parallel and Perpendicular Lines | 平行与垂直直线

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

两条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斜率的乘积为−1：m₁ × m₂ = −1（前提是两条直线均非垂直）。

For instance, y = 2x + 1 and y = 2x − 4 are parallel. The line y = −½x + 3 is perpendicular to y = 2x + 1 because 2 × (−½) = −1.

例如，y = 2x + 1与y = 2x − 4平行。y = −½x + 3与y = 2x + 1垂直，因为2 × (−½) = −1。

6. Intersection of Lines | 直线的交点

To find the intersection point of two lines, solve their equations simultaneously. The solution gives the coordinates (x, y) that lie on both lines.

求两条直线的交点就是联立求解它们的方程，解出的(x, y)同时满足两条直线。

Example: y = 2x + 1 and y = −x + 4. Equating gives 2x + 1 = −x + 4, so 3x = 3, x = 1. Then y = 3. The intersection is (1, 3).

例：y = 2x + 1与y = −x + 4。令2x + 1 = −x + 4，得3x = 3，x = 1。于是y = 3。交点为(1, 3)。

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

For a line written in general form ax + by + d = 0 and a point P(x₀, y₀), the perpendicular distance from P to the line is:

对于一般式ax + by + d = 0表示的直线和点P(x₀, y₀)，点P到直线的垂直距离为：

D = |a x₀ + b y₀ + d| / √(a² + b²)

This formula is useful for finding altitudes of triangles and computing areas using the perpendicular height.

该公式常用于求三角形的高，从而利用垂直高度计算面积。

8. Equation of a Circle | 圆的方程

In the coordinate plane, a circle with centre C(h, k) and radius r has the equation:

在坐标平面中，圆心为C(h, k)、半径为r的圆方程为：

(x − h)² + (y − k)² = r²

If the centre is at the origin, the equation simplifies to x² + y² = r². Expanding the standard form gives the general quadratic form x² + y² + 2gx + 2fy + c = 0, whose centre is (−g, −f) and radius is √(g² + f² − c).

若圆心在原点，方程简化为x² + y² = r²。将标准式展开可得一般二次形式x² + y² + 2gx + 2fy + c = 0，其圆心为(−g, −f)，半径为√(g² + f² − c)。

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

To find where a line intersects a circle, substitute the line equation into the circle equation. This produces a quadratic in one variable. The discriminant Δ can be used to determine the number of intersection points:

求直线与圆的交点时，将直线方程代入圆方程，得到关于一个变量的二次方程。利用判别式Δ可判断交点个数：

Δ > 0: two distinct intersection points – the line cuts the circle.

Δ > 0：两个不同交点——直线与圆相割。
Δ = 0: exactly one intersection point – the line is tangent to the circle.

Δ = 0：恰好一个交点——直线与圆相切。
Δ < 0: no real intersection – the line misses the circle.

Δ < 0：没有实数交点——直线与圆相离。

For example, consider the circle x² + y² = 25 and the line y = x + 1. Substituting gives x² + (x+1)² = 25 ⇒ 2x² + 2x − 24 = 0 ⇒ x² + x − 12 = 0. The discriminant is 1 + 48 = 49 > 0, so there are two intersections.

例如，圆x² + y² = 25与直线y = x + 1。代入得x² + (x+1)² = 25 ⇒ 2x² + 2x − 24 = 0 ⇒ x² + x − 12 = 0。判别式为1 + 48 = 49 > 0，所以有两个交点。

10. Area of a Triangle in Coordinate Geometry | 坐标法求三角形面积

The area of a triangle with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) can be computed directly using the determinant formula:

顶点为A(x₁, y₁), B(x₂, y₂), C(x₃, y₃)的三角形面积可以直接用行列式公式计算：

Area = ½ | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) |

Alternatively, the shoelace formula sums cross products. This method is especially useful when coordinates are not simple or when calculating polygon areas.

也可以使用鞋带公式对叉积求和。当坐标不简单或需要计算多边形面积时，这一方法尤其有用。

Example: For A(0,0), B(4,0), C(0,3), Area = ½ |0(0−3) + 4(3−0) + 0(0−0)| = ½ × 12 = 6.

例：A(0,0), B(4,0), C(0,3)，面积 = ½ |0(0−3) + 4(3−0) + 0(0−0)| = ½ × 12 = 6。

11. Locus and Basic Shapes | 轨迹与基本图形

A locus is a set of points satisfying a given condition. The most common loci in coordinate geometry are lines and circles. For example, the set of points whose distance from a fixed point (h, k) is constantly equal to r is a circle, giving the equation above.

轨迹是满足给定条件的点的集合。坐标几何中最常见的轨迹是直线和圆。例如，到定点(h, k)的距离恒等于r的点的集合是圆，其方程如上。

Another important locus is the perpendicular bisector of a segment: the set of points equidistant from the two endpoints. Its equation can be found by setting the squared distances equal.

另一个重要轨迹是线段的垂直平分线：到线段两端点距离相等的点的集合。可通过令到两端点距离的平方相等来求其方程。

12. Useful Strategy: Complete the Square | 实用技巧：配方法

When a circle equation is given in expanded form, complete the square in x and y to return to the standard (centre-radius) form.

当圆方程以展开形式给出时，分别对x和y进行配方，即可还原为标准（圆心-半径）形式。

For example, x² + y² + 6x − 4y − 12 = 0 becomes (x + 3)² − 9 + (y − 2)² − 4 − 12 = 0, hence (x + 3)² + (y − 2)² = 25. The centre is (−3, 2) and the radius is 5.

例如，x² + y² + 6x − 4y − 12 = 0可化为(x + 3)² − 9 + (y − 2)² − 4 − 12 = 0，即(x + 3)² + (y − 2)² = 25。圆心为(−3, 2)，半径为5。

Mastering these coordinate geometry tools will help you solve a wide range of IB exam problems efficiently and accurately.

掌握这些坐标几何工具，将助你在IB考试中高效、准确地解决各类问题。

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Coordinate Geometry in the Plane | 平面坐标系中的图形与计算