📚 Mastering Exercise 9B.3: Equation of a Circle | 掌握练习 9B.3：圆的方程
Exercise 9B.3 is a classic IB Mathematics problem set that explores the equation of a circle. Whether you are working through a textbook or revising for exams, understanding how to find the centre, radius, and related geometric properties is essential. This article breaks down every skill needed to tackle similar exercises with confidence and precision.
练习 9B.3 是 IB 数学中一组经典的关于圆的方程的问题。无论你是在做教材练习还是备考复习，掌握如何求圆心、半径以及相关几何性质都至关重要。本文将分解掌握这类题目所需的全部技能，让你自信、精准地解答同类问题。
1. Understanding the Standard Form | 理解圆的标准方程
The most fundamental form of a circle' e, (h, k) represents the coordinates of the centre, and r is the radius. Any point (x, y) satisfying this equation lies on the circumference.
圆的最基本方程形式为 (x − h)² + (y − k)² = r²。其中 (h, k) 表示圆心的坐标，r 为半径。任何满足该方程的点 (x, y) 都位于圆周上。
². Recognising this structure allows you to immediately extract the centre and radius without additional rearrangement.
如果圆心在原点，方程可简化为 x² + y² = r²。识别这种结构可以让你无需额外变形就能立即得出圆心和半径。
2. Converting to General Form | 化为一般式
The general form of a circle' expanded version appears frequently in IB exam questions. To return to the standard form, you must complete the square for both x and y terms.
圆的一般方程为 x² + y² + Dx + Ey + F = 0。这种展开式在 IB 考试题中出现频繁。要变回标准式，必须对 x 项和 y 项分别进行配方。
Always check that the coefficients of x² and y² are equal (usually 1). If they are not, the equation may not represent a circle or may need scaling first.
务必检查 x² 和 y² 的系数是否相等（通常为 1）。如果不相等，该方程可能不表示圆，或者需要先进行缩放。
3. Identifying the Centre and Radius | 求圆心和半径
the radius is r. Be careful with the signs: (x + 3)² corresponds to h = −3, so the centre's x-coordinate is −3.
已知标准式 (x − h)² + (y − k)² = r²，圆心为 (h, k)，半径为 r。注意符号：(x + 3)² 对应 h = −3，因此圆心的 x 坐标为 −3。
| Example Equation | 示例方程 | Centre | 圆心 | Radius | 半径 |
|---|---|---|
| (x − 2)² + (y + 4)² = 25 | (2, −4) | 5 |
| x² + (y − 1)² = 3 | (0, 1) | √3 |
4. Completing the Square Technique | 配方法
up x and y terms. Then add the square of half the linear coefficient inside each group, balancing the equation accordingly.
要将一般式 x² + y² + Dx + Ey + F = 0 化为标准式，首先把 x 项和 y 项分别分组。然后在各组内加上一次项系数一半的平方，并相应平衡等式。
For example, x² + 6x becomes (x + 3)² − 9. Repeat for the y terms, then move constants to the right side. The right side must be positive for a real circle.
例如，x² + 6x 变为 (x + 3)² − 9。对 y 项同样处理，然后将常数移到等式右边。等式右边必须为正值，才能表示一个实数圆。
5. Points on a Circle | 圆上的点
To verify whether a point (x₁, y₁) lies on a given circle, substitute its coordinates into the circle's equation. If the left side equals the right side, the point is on the circumference.
要验证点 (x₁, y₁) 是否位于给定的圆上，将其坐标代入圆的方程。若左边等于右边，则该点在圆周上。
If the left side is less than r², the point lies inside the circle. If greater, it lies outside. This distance-based reasoning is often tested in multiple-choice questions.
若左边小于 r²，该点在圆内；若大于 r²，则在圆外。这种基于距离的判断经常出现在选择题中。
6. Equation from Centre and Point | 已知圆心和一点求圆的方程
ance formula to find r² = (x₁ − h)² + (y₁ − k)². Then write the equation in standard form.
当已知圆心 (h, k) 和圆上一点 (x₁, y₁)，可利用距离公式求得 r² = (x₁ − h)² + (y₁ − k)²。然后写出标准方程。
This method also applies when you know the endpoints of a diameter: the centre is the midpoint, and the radius is half the distance between them.
若已知直径的两个端点，同样可用此方法：圆心为两端点的中点，半径为其距离的一半。
7. Intersection of Lines and Circles | 直线与圆的交点
here a line intersects a circle, solve their equations simultaneously. Substitute the line's equation y = mx + c into the circle' uation, resulting in a quadratic. The discriminant determines the number of intersections.
求直线与圆的交点时，需要联立求解两者的方程。将直线方程 y = mx + c 代入圆的方程，得到一个一元二次方程。判别式决定了交点的个数。
Δ > 0 → two intersections
Δ = 0 → one tangent point (line is tangent)
Δ < 0 → no intersection
判别式 Δ > 0 → 两个交点
Δ = 0 → 一个切点（直线为切线）
Δ < 0 → 无交点
8. Tangent to a Circle | 圆的切线
The tangent at a point on the circle is perpendicular to the radius at that point. To find the equation of the tangent, first determine the gradient of the radius, then use the negative reciprocal for the tangent's gradient.
圆上一点的切线与过该点的半径垂直。要求切线的方程，首先确定半径的斜率，然后取负倒数作为切线的斜率。
A common exam task gives an external point and asks for the equations of the two tangents from that point to the circle. This involves using the condition of tangency (distance from centre to line equals radius) with the line equation in general form.
常见的考试题型是给出一个圆外点，求从该点引出的两条切线方程。这需要利用相切条件（圆心到直线的距离等于半径），并结合直线的一般式进行求解。
9. Perpendicular Bisector of Chords | 弦的垂直平分线
A chord of a circle has its perpendicular bisector passing through the centre. This property can be used to find the centre when the circle's equation is unknown but chords are given.
圆的弦的垂直平分线会经过圆心。当圆的方程未知但给定弦时，可利用这一性质求出圆心。
If two chords are known, the intersection of their perpendicular bisectors gives the centre. Then the radius is found by measuring the distance to any chord endpoint.
若已知两条弦，它们垂直平分线的交点即为圆心。然后通过测量圆心到任一弦端点的距离即可得到半径。
10. Circle through Three Points | 过三点的圆
his yields three simultaneous linear equations in D, E, F. Solve to find the coefficients.
求经过三个不共线点的圆的方程时，将每个点代入一般式 x² + y² + Dx + Ey + F = 0，得到关于 D、E、F 的三个线性方程。解此方程组即可求出系数。
An alternative geometric method involves finding the circumcentre of the triangle formed by the points, using perpendicular bisectors. Both approaches are valid and often tested in IB.
另一种几何方法是利用垂直平分线求出三点构成三角形的外心。两种方法均有效，且常在 IB 考试中出现。
11. Real-World Applications | 实际应用
Circular equations appear in physics (projectile motion paths, circular orbits), engineering (gear designs), and computer graphics (collision detection). Understanding the underlying mathematics strengthens modelling skills in IB's internal assessments.
圆的方程出现在物理学（抛体运动的轨迹、圆轨道）、工程学（齿轮设计）以及计算机图形学（碰撞检测）中。理解背后的数学原理能增强你在 IB 内部评估中的建模能力。
ath around it can be formulated using concentric circles: x² + y² = r₁² and x² + y² = r₂².
例如，为一座圆形花园及其周围小路建模，可以使用同心圆：x² + y² = r₁² 和 x² + y² = r₂²。
12. Key Tips for IB Exams | IB 考试关键提示
- Always write the standard form clearly before extracting features; avoid working mentally with the expanded form.
- 记住在提取特征前先清晰地写出标准式；避免心算展开式。
- Check that r² is positive. If it equals zero, the equation represents a point (degenerate circle). If negative, there is no real locus.
- 检查 r² 是否为正值。若等于零，方程表示一个点（退化圆）；若为负值，则不存在真实的轨迹。
- When dealing with tangents, sketch the circle and line. This often reveals simple geometric shortcuts.
- 处理切线问题时，画出圆和直线的草图。这常常能揭示简单的几何捷径。
- Use dimensional analysis in word problems: ensure the radius has the correct unit before squaring.
- 在应用题中进行量纲分析：确保半径在平方之前具有正确的单位。
Mastering Exercise 9B.3 builds a solid foundation for advanced coordinate geometry and calculus topics. Regular practice with varied problems will make the process automatic.
掌握练习 9B.3 能为进阶坐标几何和微积分专题打下坚实基础。通过多样化的习题进行经常性练习，将使这些步骤变成自然而然的反应。
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