Circles | 圆

📚 Circles | 圆

In A-Level Edexcel Mathematics, the topic of circles is an essential part of coordinate geometry. You will learn to work with the standard equation of a circle, convert between forms, find the centre and radius, and solve problems involving intersections with lines, tangents, and chords. Mastering circles lays a solid foundation for further pure mathematics and applied contexts.

在 A-Level Edexcel 数学中，圆是坐标几何的重要组成部分。你将学习圆的标准方程、不同形式之间的转换、寻找圆心和半径，以及解决与直线交点、切线、弦相关的问题。掌握好圆的知识为进一步学习纯数学和应用背景打下坚实基础。

1. Standard Equation of a Circle | 圆的标准方程

A circle is defined as the set of all points in a plane that are a fixed distance (radius) from a fixed point (centre). If the centre is at the point (a, b) and the radius is r, the standard equation is:

圆定义为平面上到定点（圆心）距离等于定长（半径）的所有点的集合。如果圆心在点 (a, b)，半径为 r，则标准方程为：

(x – a)² + (y – b)² = r²

The derivation comes from the distance formula: the distance between any point (x, y) on the circle and (a, b) is √[(x – a)² + (y – b)²] = r. Squaring both sides gives the equation above.

推导来自距离公式：圆上任意点 (x, y) 与圆心 (a, b) 的距离为 √[(x – a)² + (y – b)²] = r，两边平方即得上述方程。

For example, the circle with centre (2, -3) and radius 5 has equation (x – 2)² + (y + 3)² = 25.

例如，圆心为 (2, -3)、半径为 5 的圆的方程为 (x – 2)² + (y + 3)² = 25。

2. Finding Centre and Radius | 求圆心和半径

Given an equation in the form (x – a)² + (y – b)² = r², you can simply read off the centre (a, b) and radius r. For example, (x + 4)² + (y – 1)² = 9 has centre (-4, 1) and radius 3.

给定形如 (x – a)² + (y – b)² = r² 的方程，你可以直接读出圆心 (a, b) 和半径 r。例如 (x + 4)² + (y – 1)² = 9 的圆心为 (-4, 1)，半径为 3。

If the equation is given in expanded form, you need to complete the square for both x and y terms to bring it to standard form.

如果方程以展开形式给出，你需要对 x 和 y 项分别配方，将其化为标准形式。

Example: x² + y² – 6x + 10y + 18 = 0. Group x terms: (x² – 6x) and y terms: (y² + 10y). Complete the square: (x – 3)² – 9 + (y + 5)² – 25 + 18 = 0, which simplifies to (x – 3)² + (y + 5)² = 16. So centre (3, -5), radius 4.

示例：x² + y² – 6x + 10y + 18 = 0。将 x 项组合：(x² – 6x)，y 项组合：(y² + 10y)。配方：(x – 3)² – 9 + (y + 5)² – 25 + 18 = 0，化简得 (x – 3)² + (y + 5)² = 16。因此圆心 (3, -5)，半径 4。

3. General Equation of a Circle | 圆的一般方程

The general equation of a circle is x² + y² + 2gx + 2fy + c = 0. This form is often used in Edexcel exam questions.

圆的一般方程为 x² + y² + 2gx + 2fy + c = 0。这种形式在 Edexcel 考试题中经常出现。

x² + y² + 2gx + 2fy + c = 0

Its centre is (-g, -f) and radius is √(g² + f² – c), provided g² + f² – c > 0.

其圆心为 (-g, -f)，半径为 √(g² + f² – c)，要求 g² + f² – c > 0。

To derive this, complete the square: (x + g)² + (y + f)² = g² + f² – c. Comparing with the standard form gives centre (-g, -f) and r = √(g² + f² – c).

推导过程：配方后得 (x + g)² + (y + f)² = g² + f² – c。与标准形式比较得圆心 (-g, -f)，半径 r = √(g² + f² – c)。

It is important to check the condition g² + f² – c > 0 for a real circle; if equal to 0, it represents a single point; if less than 0, no real circle exists.

注意检查条件 g² + f² – c > 0 以确认是一个真实的圆；如果等于 0，则代表一个点；如果小于 0，则没有实数圆存在。

4. Determining the Equation from Given Conditions | 根据条件确定圆的方程

You may be asked to find the equation of a circle that satisfies certain geometric conditions, such as passing through given points, having a given centre, or being tangent to a line. The general approach is to use the standard form (x – a)² + (y – b)² = r² and substitute known information to find a, b, and r.

你可能需要求满足特定几何条件的圆的方程，例如经过给定点、给定圆心或与某直线相切。

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