Equation of a Circle | 圆的方程

📚 Equation of a Circle | 圆的方程

In coordinate geometry, the equation of a circle is a fundamental concept that builds on the distance formula. A circle is defined as the set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the centre). Understanding how to write and interpret circle equations in both standard and general forms is crucial for A-Level mathematics, especially for solving problems involving tangents, intersections and geometric conditions.

在坐标几何中，圆的方程是建立在距离公式基础上的重要概念。圆定义为平面上到定点（圆心）的距离等于定长（半径）的所有点的集合。掌握如何写出并理解圆的标准式和一般式方程对A-Level数学至关重要，尤其在求解涉及切线、交点以及几何条件的问题时。

1. The Standard Circle Equation | 圆的标准方程

A circle with centre (a, b) and radius r has the standard equation (x − a)² + (y − b)² = r². This comes directly from applying Pythagoras’ theorem: the distance from any point (x, y) on the circle to the centre is r, so √[(x − a)² + (y − b)²] = r. Squaring both sides gives the standard form.

圆心为 (a, b)、半径为 r 的圆的标准方程为 (x − a)² + (y − b)² = r²。这直接源自勾股定理：圆上任意一点 (x, y) 到圆心的距离为 r，因此 √[(x − a)² + (y − b)²] = r。两边平方即得标准式。

For example, a circle centred at (2, −1) with radius 5 has equation (x − 2)² + (y + 1)² = 25. Note that y − (−1) becomes y + 1.

例如，以 (2, −1) 为圆心、半径为 5 的圆的方程为 (x − 2)² + (y + 1)² = 25。注意 y − (−1) 写作 y + 1。

2. Identifying Centre and Radius | 确定圆心与半径

When you are given an equation in standard form, you can read the centre and radius directly. For (x + 3)² + (y − 2)² = 16, rewrite x + 3 as x − (−3), so the centre is (−3, 2). The right side is 16, so r = √16 = 4.

当你得到标准式方程时，可直接读出圆心与半径。对于 (x + 3)² + (y − 2)² = 16，将 x + 3 改写为 x − (−3)，因此圆心为 (−3, 2)。等号右侧是 16，故半径 r = √16 = 4。

If the equation is (x − 0)² + (y − 0)² = 10, the centre is the origin (0,0) and radius is √10. The presence of zero shifts simplifies many problems.

若方程为 (x − 0)² + (y − 0)² = 10，圆心为原点 (0,0)，半径为 √10。零平移会简化许多问题。

3. Drawing Circles on the Coordinate Plane | 在坐标平面上画圆

To sketch a circle, mark the centre and then plot points r units left, right, up and down from the centre. Then draw the smooth curve through these points. For (x − 1)² + (y + 2

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