Circle Formulas: Equation, Tangents, Chords, and More | 圆公式：方程、切线、弦等

📚 Circle Formulas: Equation, Tangents, Chords, and More | 圆公式：方程、切线、弦等

In IB Mathematics, circles appear frequently in coordinate geometry, requiring a strong grasp of their equations, tangents, chords, and intersections. This article systematically presents the essential formulas and problem-solving strategies you need, with paired English–Chinese explanations to reinforce understanding.

在IB数学中，圆经常出现在坐标几何里，要求学生熟练掌握其方程、切线、弦以及交点等问题。本文系统地呈现必备公式和解题策略，并采用英中对照的方式加深理解。

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

A circle with centre C(h, k) and radius r is described by the standard equation:

(x − h)² + (y − k)² = r²

This form directly reveals the centre (h, k) and the radius r. For example, (x − 2)² + (y + 3)² = 25 represents a circle with centre (2, −3) and radius 5. If the centre is at the origin O(0, 0), the equation simplifies to x² + y² = r², a case frequently used in trigonometric definitions and the unit circle where r = 1.

圆心为C(h, k)、半径为r的圆的标准方程为 (x − h)² + (y − k)² = r²。该形式直接给出圆心(h, k)和半径r。例如 (x − 2)² + (y + 3)² = 25 表示圆心 (2, −3)、半径为5的圆。当圆心在原点O(0, 0) 时，方程简化为 x² + y² = r²，这种情况在三角函数定义和单位圆 (r = 1) 中很常用。

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2. General Form of a Circle | 圆的一般方程

Expanding the standard equation yields the general form:

x² + y² + Dx + Ey + F = 0

For this to represent a real circle (non‑degenerate), the coefficients must satisfy D² + E² − 4F > 0. If D² + E² − 4F = 0, it represents a single point; if < 0, no real graph exists. Starting from the general form, we can recover the centre and radius by completing the square or using direct formulas:

Centre (−D/2, −E/2)    Radius = ½√(D² + E² − 4F)

将标准方程展开即可得到一般式 x² + y² + Dx + Ey + F = 0。要使其表示一个真实的圆（非退化），系数必须满足 D² + E² − 4F > 0。若该式等于0，则表示一个点；若小于0，则没有实图形。从一般式出发，可以通过配方或直接使用公式求出圆心与半径：圆心 (−D/2, −E/2)，半径 = ½√(D² + E² − 4F)。

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3. Converting Between Forms by Completing the Square | 用配方法在形式之间转换

To find the centre and radius from x² + y² + Dx + Ey + F = 0, group x‑terms and y‑terms, then complete the square. Example: x² + y² − 6x + 8y + 9 = 0.

Rewrite as (x² − 6x) + (y² + 8y) = −9. Complete the squares: (x² − 6x + 9) + (y² + 8y + 16) = −9 + 9 + 16 → (x − 3)² + (y + 4)² = 16. Hence centre (3, −4), radius 4.

要从一般式 x² + y² + Dx + Ey + F = 0 求出圆心和半径，先将 x 项和 y 项分别组合，然后配方。例如：x² + y² − 6x + 8y + 9 = 0。改写为 (x² − 6x) + (y² + 8y) = −9。配方得 (x − 3)² + (y + 4)² = 16。因此圆心为 (3, −4)，半径为4。这一技巧在考试中非常重要，因为它直接给出几何信息。

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4. Parametric Form of a Circle | 圆的参数方程

A circle with centre (h, k) and radius r can be expressed in parametric form using a parameter θ (often representing the angle measured from the positive x‑direction):

x = h + r cos θ    y = k + r sin θ    (0 ≤ θ < 2π)

This representation is especially useful when describing motion along the circle, converting trigonometric problems into coordinate geometry, and when using calculus to find slopes of tangents via dy/dθ and dx/dθ.

圆心为(h, k)、半径为r的圆可以用参数θ（通常表示从正x方向量起的角度）写成参数形式：x = h + r cos θ, y = k + r sin θ (0 ≤ θ < 2π)。这种方法在描述圆周运动、将三角问题转化为坐标几何，以及利用微积分通过 dy/dθ 与 dx/dθ 求切线斜率时非常有用。

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5. Tangent to a Circle at a Given Point | 圆上一点处的切线

A tangent to a circle is perpendicular to the radius drawn to the point of contact. If the circle is given by (x − h)² + (y − k)² = r² and the point of tangency is T(x₁, y₁) on the circle, the tangent equation can be written directly:

(x₁ − h)(x − h) + (y₁ − k)(y − k) = r²

For a circle in general form x² + y² + Dx + Ey + F = 0, the tangent at (x₁, y₁) is:

xx₁ + yy₁ + D(x + x₁)/2 + E(y + y₁)/2 + F = 0

These formulas arise from replacing x² with xx₁, y² with yy₁, x with (x + x₁)/2, and y with (y + y₁)/2. Alternatively, you can find the gradient of the radius and take the negative reciprocal to obtain the tangent slope.

圆的切线垂直于过切点的半径。若圆为 (x − h)² + (y − k)² = r²，切点为圆上的点 T(x₁, y₁)，则切线方程可直接写为 (x₁ − h)(x − h) + (y₁ − k)(y − k) = r²。对于一般式 x² + y² + Dx + Ey + F = 0，在 (x₁, y₁) 处的切线方程为 xx₁ + yy₁ + D(x + x₁)/2 + E(y + y₁)/2 + F = 0。这些公式来源于用 xx₁ 替换 x²、用 yy₁ 替换 y²、用 (x + x₁)/2 替换 x、用 (y + y₁)/2 替换 y。此外，也可以求出半径的斜率，取其负倒数得到切线斜率。

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6. Chord of a Circle and Its Properties | 圆的弦及其性质

A chord is a line segment whose endpoints lie on the circle. The perpendicular from the centre of a circle to a chord bisects the chord. This fundamental property is often used to solve problems involving chord length and distance from the centre.

Using Pythagoras’ theorem: if the distance from the centre to a chord is d, and the radius is r, then half the chord length is √(r² − d²), so the full chord length L is:

L = 2√(r² − d²)

Given the equation of a line and a circle, you can find the chord endpoints by solving the system, but the length is more efficiently computed via the perpendicular distance from the centre to the line.

弦是端点都在圆上的线段。从圆心到弦的垂线平分该弦。这一基本性质常用于解决与弦长和圆心到弦的距离有关的问题。利用勾股定理：若圆心到弦的距离为 d，半径为 r，则弦的一半为 √(r² − d²)，因此整个弦长 L = 2√(r² − d²)。给定一条直线和一个圆的方程，可以通过解方程组求出弦的端点，但利用圆心到直线的垂距来计算弦长更为高效。

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7. Intersection of a Line and a Circle | 直线与圆的交点

To determine the intersection between a line y = mx + c and a circle (x − h)² + (y − k)² = r², substitute the line equation into the circle equation. This produces a quadratic in x (or y). The discriminant Δ of this quadratic reveals the nature of the intersection:

• Δ > 0: two distinct intersection points (secant line)
• Δ = 0: one point of tangency (tangent line)
• Δ < 0: no real intersection (line outside the circle)

The discriminant method is extremely useful when you need to find conditions for a line to be tangent (e.g., to find values of c for which y = mx + c is tangent).

要确定直线 y = mx + c 与圆 (x − h)² + (y − k)² = r² 的交点情况，可将直线方程代入圆的方程，从而得到一个关于 x（或 y）的二次方程。该二次方程的判别式 Δ 揭示了交点的性质：Δ > 0 有两个不同交点（割线），Δ = 0 有一个切点（切线），Δ < 0 没有实交点（直线在圆外）。当需要找出直线为切线的条件时（例如求 y = mx + c 相切时的 c 值），判别式法非常有用。

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8. Tangents from an External Point | 从圆外一点引圆的切线

Given a circle (x − h)² + (y − k)² = r² and an external point P(x₀, y₀), two tangents can be drawn to the circle. The length of the tangent segment from P to the point of contact is given by:

PT = √((x₀ − h)² + (y₀ − k)² − r²)

This formula comes from the power of a point and is essentially the square root of substituting P into the circle’s equation in standard form (or into x² + y² + Dx + Ey + F for the general form). To find the equations of the two tangents, one common approach is to let the tangent have slope m, write its equation as y − y₀ = m(x − x₀), and then set the perpendicular distance from the centre to this line equal to r. Solving for m yields two values, giving both tangents.

给定圆 (x − h)² + (y − k)² = r² 和一个圆外点 P(x₀, y₀)，可以作两条切线。从 P 到切点的切线长度由 PT = √((x₀ − h)² + (y₀ − k)² − r²) 给出。该公式源自点幂，本质上是将 P 代入标准式（或一般式 x² + y² + Dx + Ey + F）后取平方根。为求出两条切线的方程，常用方法是设切线斜率为 m，写出点斜式 y − y₀ = m(x − x₀)，然后利用圆心到该直线的垂距等于 r 的条件解出 m，从而得到两条切线。

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9. Power of a Point and Secant–Tangent Theorem | 点幂与割线–切线定理

For a circle and a point P, the power of P is defined as d² − r², where d is the distance from P to the centre. This value is positive outside, zero on, and negative inside the circle.

If a line through P intersects the circle at A and B, then the product PA × PB is constant for all such lines and equals |d² − r²| (taking directed segments, it equals d² − r²). In particular, for a secant and a tangent from P, if PT is the tangent length and PAB is a secant, we have:

PT² = PA × PB

This result is often called the tangent–secant theorem. When two secants PAB and PCD are drawn, PA × PB = PC × PD. These relationships are powerful shortcuts in geometry problems involving intersecting chords and secants.

对于一个圆和一点 P，点 P 的幂定义为 d² − r²，其中 d 是 P 到圆心的距离。该值在圆外为正，在圆上为零，在圆内为负。若过 P 的任一直线与圆交于 A、B，则乘积 PA × PB 对所有这样的直线都是常数，且等于 |d² − r²|（若使用有向线段，则等于 d² − r²）。特别地，对于从 P 出发的一条切线和一条割线，若 PT 为切线长，PAB 为割线，则有 PT² = PA × PB。这一结果常被称为切线–割线定理。当有两条割线 PAB 和 PCD 时，有 PA × PB = PC × PD。这些关系式在涉及相交弦和割线的几何问题中是强大的捷径。

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10. Equation of a Circle Through Three Non‑Collinear Points | 过三个不共线点的圆的方程

Three non‑collinear points uniquely determine a circle. One method is to assume the general form x² + y² + Dx + Ey + F = 0, substitute the coordinates of the three points, and solve the resulting 3×3 linear system for D, E, F. An alternative approach uses the perpendicular bisectors of two chords: their intersection gives the centre, and the radius is the distance from the centre to any of the three points.

Example: Find the circle through A(1, 0), B(3, 2), C(5, 0). By substituting into the general form, you obtain three equations. Solving gives centre (3, 0) and radius 2, hence the circle (x − 3)² + y² = 4. In coordinate geometry problems, the algebraic method with the general form is systematic and reliable.

三个不共线的点唯一确定一个圆。一种方法是设一般式 x² + y² + Dx + Ey + F = 0，代入三个点的坐标，解出关于 D、E、F 的 3×3 线性方程组。另一种方法是利用两条弦的垂直平分线：它们的交点即为圆心，半径则是圆心到任一点的距离。例如：求过 A(1, 0)、B(3, 2)、C(5, 0) 的圆。代入一般式得到三个方程，求解得圆心 (3, 0)、半径2，因此圆的方程为 (x − 3)² + y² = 4。在坐标几何问题中，使用一般式的代数方法系统且可靠。

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11. Summary of Essential Circle Formulas | 圆的关键公式总结

Description	Formula
Standard equation	(x − h)² + (y − k)² = r²
General form	x² + y² + Dx + Ey + F = 0
Centre from general form	(−D/2, −E/2)
Radius from general form	½√(D² + E² − 4F)
Parametric form	x = h + r cos θ, y = k + r sin θ
Tangent at (x₁, y₁) (standard form)	(x₁ − h)(x − h) + (y₁ − k)(y − k) = r²
Tangent at (x₁, y₁) (general form)	xx₁ + yy₁ + D(x + x₁)/2 + E(y + y₁)/2 + F = 0
Chord length (distance d from centre)	L = 2√(r² − d²)
Tangent length from external point (x₀, y₀)	√((x₀ − h)² + (y₀ − k)² − r²)
Condition for line y = mx + c to be tangent	Distance from centre to line = r

This table provides a quick reference for solving typical IB Mathematics problems on circles. Alongside understanding these formulas, practising their application in varied contexts—such as finding the equation of a tangent, calculating chord lengths, and working with the power of a point—will build the fluency expected in exams.

此表格为求解典型IB数学圆问题提供了快速参考。除了理解这些公式外，在多种情境中练习它们的应用——例如求切线方程、计算弦长、运用点幂——将培养考试中所期望的熟练度。

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