📚 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。
al 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
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。
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。
g. 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
he 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。
| Discriminant Δ 判别式 Δ |
Number of Intersections 交点个数 |
Geometric Meaning 几何意义 |
| Δ > 0 | Two distinct points 两个不同交点 |
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