Intersection of Lines and Circles | 直线与圆的交点问题

📚 Intersection of Lines and Circles | 直线与圆的交点问题

The intersection of a straight line with a circle is a cornerstone of coordinate geometry in Edexcel A-Level Mathematics. This topic links algebraic manipulation with geometric intuition, and examiners use it to test whether you can move fluently between equations, graphs, and conditions such as tangency.

直线与圆的交点是Edexcel A-Level数学中坐标几何的基石。这一专题将代数运算与几何直觉紧密结合，考官常用它来检验你是否能在方程、图形以及相切等条件之间灵活转换。

1. The Standard Equations You Must Know | 必备的标准方程

Before attempting any intersection problem, you need complete fluency with the two forms of a circle’s equation and the two main forms of a straight line.

在尝试任何交点问题之前，你必须完全熟练圆的方程的两种形式以及直线的两种主要形式。

A circle with centre (a, b) and radius r is written in centre-radius form as follows.

圆心为 (a, b)、半径为 r 的圆，其圆心-半径形式如下。

(x − a)² + (y − b)² = r²

Expanding the brackets produces the general form of a circle:

展开括号后得到圆的一般式：

x² + y² + 2gx + 2fy + c = 0

This general form has centre (−g, −f) and radius √(g² + f² − c). You can switch between the two forms by completing the square.

该一般式的圆心为 (−g, −f)，半径为 √(g² + f² − c)。你可以在两种形式之间通过配方互相转换。

The line used in these problems is normally given either in gradient form y = mx + c, or in general form ax + by + c = 0. Both are useful, and you should be ready to rearrange either one.

此类问题中的直线通常以斜截式 y = mx + c 或一般式 ax + by + c = 0 给出。两者都很有用，你应该随时准备对它们进行变形。

2. The Substitution Method | 代入法

To find where a line meets a circle, substitute the equation of the line into the equation of the circle. If the line is y = mx + c, replace every y in the circle equation with mx + c.

要求直线与圆的交点，方法是将直线方程代入圆的方程。若直线为 y = mx + c，则将圆的方程中所有 y 替换为 mx + c。

Using the centre-radius form as an example, substituting y = mx + c into (x − a)² + (y − b)² = r² gives a quadratic in x.

以圆心-半径形式为例，将 y = mx + c 代入 (x − a)² + (y − b)² = r² 后，得到一个关于 x 的一元二次方程。

Ax² + Bx + C = 0

This quadratic is the key to the entire problem. Its discriminant tells you the number of intersection points, and its roots give you the x-coordinates of those points.

这个二次方程是整个问题的关键。它的判别式告诉你交点的个数，而它的根给出这些交点的 x 坐标。

If the line is given in general form ax + by + c = 0, rearrange it to make y the subject (or x, if that is simpler) before substituting. For a vertical line such as x = k, simply plug x = k directly into the circle equation.

如果直线以一般式 ax + by + c = 0 给出，先将其变形为 y 的表达式（若更方便也可解出 x）再代入。对于竖直线如 x = k，直接把 x = k 代入圆的方程即可。

3. The Discriminant: Your Key Tool | 判别式：核心工具

For a quadratic Ax² + Bx + C = 0, the discriminant is defined as Δ = B² − 4AC. In the context of line-circle intersections, this value completely determines how many points the line and circle share.

对于一元二次方程 Ax² + Bx + C = 0，判别式定义为 Δ = B² − 4AC。在直线与圆的交点问题中，该值完全决定了直线与圆共有多少个点。

The three cases are summarised in the table below.

三种情况总结如下表。