📚 Exercise 6D: Coordinate Geometry of Circles | 练习 6D：圆的坐标几何
Exercise 6D in AQA Pure Mathematics brings together the standard equation of a circle, completing the square, tangents, normals, chords and intersection conditions. These skills appear regularly in AS and A-Level exam questions, often mixed with straight line equations.
AQA 纯数学练习 6D 综合考察圆的标准方程、配方法、切线、法线、弦以及直线与圆的交点条件。这些技能在 AS 和 A-Level 考试中经常出现，并且经常与直线方程综合考查。
1. The Standard Equation of a Circle | 圆的标准方程
starting point for most circle problems. The centre is (a, b) and the radius is r, where r > 0.
在 AQA 考题中，标准形式 (x − a)² + (y − b)² = r² 是大多数圆问题的起点。圆心为 (a, b)，半径为 r，且要求 r > 0。
(x − a)² + (y − b)² = r²
tly from the brackets. For example, (x − 3)² + (y + 2)² = 16 has centre (3, −2) and radius √16 = 4.
你必须学会直接从括号读取圆心。例如，圆 (x − 3)² + (y + 2)² = 16 的圆心为 (3, −2)，半径为 √16 = 4。
| Form | Equation | Centre | Radius |
|---|---|---|---|
| Standard form | (x − a)² + (y − b)² = r² | (a, b) | r |
| Expanded form | x² + y² + 2gx + 2fy + c = 0 | (−g, −f) | √g² + f² − c |
Note the sign change: if the bracket contains x + 5, the x-coordinate of the centre is −5, not 5.
注意符号变化：如果括号内是 x + 5，那么圆心的 x 坐标是 −5，而不是 5。
2. Completing the Square to Find the Centre and Radius | 用配方法求圆心和半径
If a circle is given in expanded form, group the x terms and y terms, then complete the square for each group. This converts the equation back into standard form.
如果圆以一般式给出，请将 x 项和 y 项分组，然后分别对每组进行配方。这样可以将方程转换回标准形式。
Example: x² + y² − 6x + 4y − 12 = 0.
示例：x² + y² − 6x + 4y − 12 = 0。
x² − 6x + y² + 4y − 12 = 0
(x − 3)² − 9 + (y + 2)² − 4 − 12 = 0
(x − 3)² + (y + 2)² = 25
Therefore the centre is (3, −2) and the radius is √25 = 5.
因此圆心为 (3, −2)，半径为 √25 = 5。
Remember that the constant terms created by completing the square must be moved to the right-hand side. Many AQA mark schemes reward this rearrangement step.
请记住，配方后产生的常数项必须移到等式右边。AQA 的许多评分方案会给这个移项步骤分数。
3. Position of a Point Relative to a Circle | 点与圆的位置关系
To decide whether a point P(x₁, y₁) lies inside, on, or outside a circle with centre (a, b) and radius r, compare the squared distance CP² with r².
要判断点 P(x₁, y₁) 是在圆内、圆上还是圆外，可以比较平方距离 CP² 与 r²。
CP² = (x₁ − a)² + (y₁ − b)²
- If CP² < r², the point is inside the circle.
- If CP² = r², the point is on the circle.
- If CP² > r², the point is outside the circle.
如果 CP² < r²，点在圆内；如果 CP² = r²，点在圆上；如果 CP² > r²，点在圆外。
= (5 − 2)² + (3 + 1)² = 9 + 16 = 25. Since 25 > 16, the point lies outside the circle.
示例：对于圆 (x − 2)² + (y + 1)² = 16 和点 (5, 3)，CP² = (5 − 2)² + (3 + 1)² = 9 + 16 = 25。因为 25 > 16，所以点在圆外。
4. Tangents to a Circle | 圆的切线
A tangent to a circle at a point P is perpendicular to the radius drawn to P. This perpendicular relationship is the key to finding the tangent equation.
圆在点 P 处的切线与过点 P 的半径垂直。这种垂直关系是求切线方程的关键。
If the radius has gradient m₁, and m₁ ≠ 0, then the tangent has gradient m₂ given by:
如果半径的斜率为 m₁，且 m₁ ≠ 0，则切线的斜率 m₂ 为：
m₂ = −1 / m₁
If the radius is vertical, its gradient is undefined; the tangent is then horizontal. If the radius is horizontal, the tangent is vertical.
如果半径是竖直的，其斜率不存在，那么切线是水平线。如果半径是水平的，那么切线是竖直线。
nce you have the tangent gradient, use y − y₁ = m(x − x₁) with the known point of contact.
求出切线斜率后，使用已知切点代入 y − y₁ = m(x − x₁) 即可得到切线方程。
5. Normals to a Circle | 圆的法线
The normal to a circle at a point P is the line passing through P and the centre of the circle. It is therefore the same line as the radius at P.
圆在点 P 处的法线是经过点 P 和圆心的直线。因此，法线就是过点 P 的半径所在的直线。
This is a useful shortcut: do not treat the normal as an unknown line. Its gradient is simply the gradient of the radius at that point.
这是一个有用的捷径：不要把法线当作未知直线。它的斜率就是该点处半径的斜率。
the normal gradient is (8 − 4)/(4 − 1) = 4/3. The normal equation is y − 8 = (4/3)(x − 4).
例如，若圆心为 (1, 4)，P 为 (4, 8)，则法线斜率为 (8 − 4)/(4 − 1) = 4/3。法线方程为 y − 8 = (4/3)(x − 4)。
6. Chords and the Perpendicular from the Centre | 弦与圆心到弦的垂线
In any circle, the perpendicular from the centre to a chord bisects the chord. This theorem is used to find missing distances and to prove geometric properties.
在任意圆中，圆心到弦的垂线平分该弦。这个定理用于求缺失距离以及证明几何性质。
If the circle has radius r and the perpendicular distance from the centre to a chord is d, then the half-length of the chord is:
如果圆的半径为 r，圆心到弦的垂直距离为 d，则弦的一半长度为：
Half chord = √r² − d²
This formula comes directly from Pythagoras theorem. Draw the radius to one end of the chord and the perpendicular from the centre to form a right-angled triangle.
该公式直接来自勾股定理。画出圆心到弦一端的半径以及圆心到弦的垂线，即可形成一个直角三角形。
7. Intersections of Lines and Circles | 直线与圆的交点
cle, substitute the line equation into the circle equation. This produces a quadratic in x.
要求直线 y = mx + c 与圆的交点，请将直线方程代入圆的方程。这将产生一个关于 x 的二次方程。
x² + bx + c = 0, the discriminant D = b² − 4ac tells you how many intersections there are:
对于得到的二次方程 ax² + bx + c = 0，判别式 D = b² − 4ac 可以判断交点个数：
- D > 0 means the line cuts the circle at two distinct points.
- D = 0 means the line is a tangent to the circle.
- D < 0 means the line does not meet the circle.
D > 0 表示直线与圆有两个不同交点；D = 0 表示直线为圆的切线；D < 0 表示直线与圆没有交点。
Using the discriminant is faster than solving the quadratic fully, especially in AQA questions that ask you to show that a line is a tangent.
使用判别式比完整解二次方程更快
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