📚 Circles | 圆与圆的方程
In AQA A-Level Mathematics, the circle is one of the first coordinate-geometry objects that connects algebraic equations with geometric properties. This chapter shows how to write the equation of a circle, how to find its centre and radius, and how to use straight-line methods to solve problems involving intersections, tangents and chords.
在 AQA A-Level 数学中，圆是将代数与几何结合起来的重要研究对象。本章讲解如何写出圆的方程、如何求圆心和半径，以及如何使用直线方程方法解决相交、切线和弦有关的问题。
1. The Equation of a Circle | 圆的标准方程
A circle is the set of all points whose distance from a fixed point is constant. The fixed point is the centre, and the constant distance is the radius.
圆是到定点距离等于定长的所有点的集合。这个定点是圆心，定长是半径。
If the centre is (a,b)? Actually, we avoid LaTeX: if the centre is (a,b) and the radius is r, then for any point (x,y) on the circle, the distance from (x,y) to (a,b) must be r.
若圆心为 (a,b)，半径为 r，则圆周上任意一点 (x,y) 到 (a,b) 的距离都等于 r。
Using the distance formula and squaring both sides gives the standard equation of a circle.
利用距离公式并将两边平方，就得到圆的标准方程。
(x − a)² + (y − b)² = r²
For example, a circle with centre (2, −1) and radius 5 has equation:
例如，圆心为 (2, −1)、半径为 5 的圆方程为：
(x − 2)² + (y + 1)² = 25
b = −1. The signs inside the brackets are opposite to the signs of the centre coordinates.
注意当 b = −1 时，y − b 变成 y + 1。括号内的符号与圆心坐标的符号相反。
2. The General Form of a Circle | 圆的一般方程
Expanding the standard equation produces an alternative way to write a circle.
将标准方程展开，可以得到圆的另一种表示形式。
(x − a)² + (y − b)² = r² expands to x² + y² − 2ax − 2by + a² + b² − r² = 0.
(x − a)² + (y − b)² = r² 展开后为 x² + y² − 2ax − 2by + a² + b² − r² = 0。
This can be written with constants 2g, 2f and c as:
可以引入常数 2g、2f 和 c，写成：
x² + y² + 2gx + 2fy + c = 0
In this general form, the centre is (−g, −f) and the radius is
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