Equation of a Circle | 圆的方程

📚 Equation of a Circle | 圆的方程

In coordinate geometry, a circle is defined as the set of all points in a plane that are a fixed distance from a given point, called the centre. The fixed distance is the radius. The equation of a circle expresses this definition algebraically, allowing us to find the centre, radius, and intersections with other curves. This article covers the standard form, general form, completing the square, tangent conditions, and common Edexcel A-Level exam techniques.

在坐标几何中，圆被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。圆的方程将这个定义用代数形式表达出来，使我们能求出圆心、半径以及与其它曲线的交点。本文涵盖标准方程、一般方程、配方法、切线条件以及 Edexcel A-Level 常见考试技巧。

1. Standard Form of a Circle | 圆的标准方程

The standard form of the equation of a circle with centre (a, b) and radius r is:

(x – a)² + (y – b)² = r²

圆心为 (a, b)、半径为 r 的圆的标准方程就是上式。

If the centre is at the origin O(0, 0), the equation simplifies to:

x² + y² = r²

如果圆心在原点 O(0, 0)，方程简化为 x² + y² = r²。

2. Centre and Radius from Standard Form | 由标准方程求圆心与半径

When a circle is given in standard form, the centre is (a, b) and the radius is r, but the signs inside the brackets are opposite to the coordinates of the centre.

当圆以标准方程给出时，圆心为 (a, b)，半径是 r，但括号内的符号与圆心坐标相反。

For example, in (x – 3)² + (y + 2)² = 16, the centre is (3, -2), and the radius is √16 = 4.

例如，在 (x – 3)² + (y + 2)² = 16 中，圆心是 (3, -2)，半径是 √16 = 4。

Always write the radius as a positive length: if the right-hand side is r², then r = √(r²) > 0.

半径始终写成正长度：如果右边是 r²，则 r = √(r²) 大于 0。

3. General Form of a Circle | 圆的一般方程

Expanding the standard form gives the general equation of a circle:

x² + y² + 2gx + 2fy + c = 0

将标准方程展开就得到圆的一般方程：x² + y² + 2gx + 2fy + c = 0。

In this form, the centre is (-g, -f) and the radius is:

r = √(g² + f² – c)

在这种形式下，圆心为 (-g, -f)，半径为 r = √(g² + f² – c)。

This is valid only when g² + f² – c > 0. If g² + f²

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