Circles | AQA A-Level数学第6章：圆

📚 Circles | AQA A-Level数学第6章：圆

In AQA A-Level Mathematics Chapter 6, circles are studied within coordinate geometry. You need to find centres and radii, complete the square, solve intersections with lines, and use the tangent-radius property accurately.

在 AQA A-Level 数学第6章中，圆属于坐标几何的重要内容。你需要能够求圆心和半径、通过配方法化一般式为标准式、求直线与圆的交点，并准确运用切线与半径垂直的性质。

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

A circle is the set of points at a fixed distance r from a fixed centre (a, b).

圆是到定点 (a, b) 的距离等于定长 r 的所有点的集合。

(x – a)² + (y – b)² = r²

The values a and b are the x- and y-coordinates of the centre, and r is the radius.

a 和 b 分别是圆心的 x 坐标和 y 坐标，r 是半径。

If the centre is at the origin, the equation simplifies to x² + y² = r².

如果圆心在原点，方程简化为 x² + y² = r²。

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2. Centre and Radius by Inspection | 通过观察求圆心和半径

When the equation is given in standard form, the centre is (a, b) and the radius is r, not r².

当方程以标准形式给出时，圆心为 (a, b)，半径为 r，而不是 r²。

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

例如 (x – 3)² + (y + 2)² = 16，圆心为 (3, -2)，半径为 √16 = 4。

Beware: a sign change occurs inside the brackets because (x – a) means the centre has x-coordinate a, while (x + 2) means a = -2.

注意：括号内会出现符号变化，因为 (x – a) 表示圆心的 x 坐标为 a，而 (x + 2) 表示 a = -2。

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3. Completing the Square | 配方法

The expanded equation x² + y² + 2gx + 2fy + c = 0 can be rewritten by completing the square.

展开式 x² + y² + 2gx + 2fy + c = 0 可以通过配方法重写。

x² + y² + 2gx + 2fy + c = 0 → (x + g)² + (y + f)² = g² + f² – c

Complete the square on x² + y² – 6x + 4y – 12 = 0.

对 x² + y² – 6x + 4y – 12 = 0 配方。

First group x terms and y terms: (x² – 6x) + (y² + 4y) = 12, then add the required constants to both sides.

首先将 x 项和 y 项分组：(x² – 6x) + (y² + 4y) = 12，然后在等式两边加上所需常数。

(x – 3)² + (y + 2)² = 25, so the centre is (3, -2) and the radius is 5.

(x – 3)² + (y + 2)² = 25，所以圆心为 (3, -2)，半径为 5。

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4. General Form and Conditions | 一般式与条件

Compare the general form x² + y² + 2gx + 2fy + c = 0 with the completed square form to identify the centre (-g, -f) and radius √(g² + f² – c).

将一般式 x² + y² + 2gx + 2fy + c = 0 与配方式进行比较，可以确定圆心 (-g, -f) 和半径 √(g² + f² – c)。

The general form requires g² + f² – c > 0 for a real circle. If it equals 0 the circle has zero radius at a single point; if negative, no real circle exists.

一般式要求 g² + f² – c > 0 才表示实圆。若等于 0，则圆退化为一个点；若小于 0，则不存在实圆。

Form	Equation	Centre	Radius
Standard form	(x – a)² + (y – b)² = r²	(a, b)	r
Centre at origin	x² + y² = r²	(0, 0)	r
General form	x² + y² + 2gx + 2fy + c = 0	(-g, -f)	√(g² + f² – c)

Always check the sign of g² + f² – c before stating a radius in a general-form question.

在一般式问题中，写出半径前一定要检查 g² + f² – c 的符号。

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5. Points and Circles | 点与圆的位置关系

To classify a point P(x₁, y₁), substitute its coordinates into the left-hand side of the circle equation, or compare the distance from the centre to the radius.

要判断点 P(x₁, y₁) 的位置，可将其坐标代入圆的方程左边，或比较该点到圆心的距离与半径。

d² = (x₁ – a)² + (y₁ – b)²

If d² < r² the point is inside; if d² = r² it lies on the circle; if d² > r² it is outside.

若 d² < r²，点在圆内；若 d² = r²，点在圆上；若 d² > r²，点在圆外。

This comparison is useful when proving whether a point lies on a given circle or when checking a tangent point.

这种比较在证明点是否在给定圆上或检验切点时非常有用。

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6. Line and Circle Intersections | 直线与圆的交点

Substitute the line equation into the circle equation to obtain a quadratic in one variable.

将直线方程代入圆的方程，得到关于一个变量的二次方程。

Use the discriminant of the resulting quadratic to determine the number of intersections.

利用所得二次方程的判别式来确定交点个数。

Δ = b² – 4ac

If Δ > 0, there are two distinct intersection points; if Δ = 0, the line is tangent; if Δ < 0, there is no intersection.

若 Δ > 0，有两个不同的交点；若 Δ = 0，直线为切线；若 Δ < 0，没有交点。

This method is the standard approach for finding where lines cut circles and for proving tangency conditions.

这是求直线与圆交点以及证明相切条件的标准方法。

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7. Tangent-Radius Property | 切线与半径的性质

The radius from the centre to a point of contact is perpendicular to the tangent at that point.

从圆心到切点的半径与该点的切线垂直。

For a circle centre (a, b) and a point P(x₁, y₁) on the circle, the gradient of the radius is (y₁ – b) / (x₁ – a).

对于圆心为 (a, b) 且点 P(x₁, y₁) 在圆上，半径的斜率为 (y₁ – b) / (x₁ – a)。

m(radius) = (y₁ – b) / (x₁ – a), m(tangent) = -(x₁ – a) / (y₁ – b)

The tangent gradient is the negative reciprocal of the radius gradient, provided the radius is not vertical.

切线斜率是半径斜率的负倒数，前提是半径不垂直于 x 轴。

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8. Finding Tangent Equations | 求切线方程

Find the tangent to (x – 2)² + (y + 1)² = 10 at P(5, -2).

求圆 (x – 2)² + (y + 1)² = 10 在点 P(5, -2) 处的切线。

First identify the centre (2, -1). The gradient of the radius is (-2 + 1) / (5 – 2) = -1/3.

首先确定圆心 (2,

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