Standard and General Form of the Equation of a Circle | 圆的标准方程与一般方程转化

📚 Standard and General Form of the Equation of a Circle | 圆的标准方程与一般方程转化

A circle is one of the most fundamental curves in coordinate geometry. In A-Level mathematics, you will often need to convert between the standard form and the general form of a circle’s equation. This article explains both forms clearly, shows how to convert from one to the other, and highlights common exam pitfalls.

圆是坐标几何中最基本的曲线之一。在 A-Level 数学中，我们经常需要在圆的标准方程与一般方程之间进行转化。本文将清晰讲解这两种形式，演示它们之间的转化方法，并指出常见的考试易错点。

---

1. Standard Form of a Circle | 圆的标准方程

The standard form of a circle’s equation is derived directly from the definition of a circle: the set of all points at a fixed distance (the radius) from a fixed point (the centre).

圆的标准方程直接来源于圆的定义：到定点（圆心）距离等于定长（半径）的所有点的集合。

(x − a)² + (y − b)² = r²

Here, (a, b) is the centre of the circle and r is the radius. This form is extremely useful because it allows us to read the centre and radius directly from the equation.

其中 (a, b) 是圆心坐标，r 是半径。这种形式的优点在于我们可以直接从方程中读出圆心和半径。

For example, consider the equation:

例如，考虑方程：

(x − 3)² + (y + 2)² = 25

Rewrite (y + 2)² as (y − (−2))². Therefore, the centre is (3, −2) and the radius is √25 = 5.

将 (y + 2)² 改写为 (y − (−2))²。因此圆心为 (3, −2)，半径为 √25 = 5。

---

2. General Form of a Circle | 圆的一般方程

When we expand the standard form and rearrange the terms, we obtain the general form of a circle’s equation:

将标准方程展开并重新整理各项后，我们得到圆的一般方程：

x² + y² + Dx + Ey + F = 0

where D, E and F are constants. Note that the coefficients of x² and y² must both be 1 (or equal if the equation is multiplied by a constant).

其中 D、E 和 F 为常数。注意 x² 和 y² 的系数必须都为 1（若方程整体乘以一个常数，则二者系数相等）。

It is important to understand that every circle can be written in this form, but not every equation of this form represents a real circle. The condition for a real circle will be discussed in Section 3.

需要理解的是，每个圆都可以写成这种形式，但并非所有这种形式的方程都表示一个真实的圆。真实圆的条件将在第 3 节讨论。

---

3. Condition for a Real Circle | 真实圆的条件

Given the general equation x² + y² + Dx + Ey + F = 0, we can complete the square to determine when it represents a real circle.

对于一般方程 x² + y² + Dx + Ey + F = 0，我们可以通过配方法来判断它何时表示一个真实的圆。

(x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² − F

Comparing with the standard form, the centre is:

与标准方程比较，圆心为：

(−D/2, −E/2)

and the radius squared is:

半径的平方为：

r² = (D² + E²)/4 − F

Therefore, the equation represents a real circle if and only if (D² + E²)/4 − F > 0. If this value equals zero, it represents a single point; if it is negative, the equation has no real locus.

因此，该方程表示真实圆的条件是 (D² + E²)/4 − F > 0。若该值等于零，则方程表示一个点；若为负数，则方程没有实数轨迹。

---

4. Converting from Standard Form to General Form | 标准方程转化为一般方程

This conversion is straightforward: expand the brackets and simplify. No special techniques are required.

这种转化非常直接：展开括号并化简即可，不需要特殊技巧。

Let us work through an example. Convert (x − 1)² + (y + 4)² = 9 to general form.

我们来看一个例子。将 (x − 1)² + (y + 4)² = 9 化为一般方程。

Step 1: Expand the squares:

第一步：展开平方项：

(x² − 2x + 1) + (y² + 8y + 16) = 9

Step 2: Combine like terms and rearrange:

第二步：合并同类项并整理：

x² + y² − 2x + 8y + 1 + 16 − 9 = 0

x² + y² − 2x + 8y + 8 = 0

Thus, the general form is x² + y² − 2x + 8y + 8 = 0.

因此，一般方程为 x² + y² − 2x + 8y + 8 = 0。

---

5. Converting from General Form to Standard Form | 一般方程转化为标准方程

This conversion requires completing the square for both x and y terms. This is a core skill tested frequently in exams.

这种转化需要对 x 项和 y 项分别配方。这是考试中经常考查的核心技能。

Let us convert x² + y² + 6x − 10y + 18 = 0 into standard form.

我们来将 x² + y² + 6x − 10y + 18 = 0 化为标准方程。

Step 1: Rearrange, grouping x terms and y terms together, keeping the constant on the right:

第一步：重新排列，将 x 项和 y 项分别分组，常数项移到右边：

(x² + 6x) + (y² − 10y) = −18

Step 2: Complete the square for x. Take half of 6, which is 3, and square it to get 9:

第二步：对 x 配方。6 的一半是 3，其平方为 9：

(x + 3)² − 9

Step 3: Complete the square for y. Take half of −10, which is −5, and square it to get 25:

第三步：对 y 配方。−10 的一半是 −5，其平方为 25：

(y − 5)² − 25

Step 4: Substitute back into the equation:

第四步：代回原方程：

(x + 3)² − 9 + (y − 5)² − 25 = −18

(x + 3)² + (y − 5)² = 16

Therefore, the centre is (−3, 5) and the radius is √16 = 4.

因此，圆心为 (−3, 5)，半径为 √16 = 4。

---

6. Worked Examples | 典型例题精讲

Let us examine two typical exam questions involving circle equations.

我们来看两道典型的考试题目，涉及圆的方程。

Example 1 | 例题 1

A circle has centre (2, −3) and passes through the point (5, 1). Find the equation of the circle in general form.

一个圆的圆心为 (2, −3)，且经过点 (5, 1)。求该圆的一般方程。

Step 1: Find the radius using the distance formula:

第一步：用距离公式求半径：

r = √((5 − 2)² + (1 − (−3))²) = √(3² + 4²) = √25 = 5

Step 2: Write the standard form:

第二步：写出标准方程：

(x − 2)² + (y + 3)² = 25

Step 3: Expand to general form:

第三步：展开为一般方程：

x² − 4x + 4 + y² + 6y + 9 = 25

x² + y² − 4x + 6y − 12 = 0

Example 2 | 例题 2

Show that x² + y² − 8x + 2y + 1 = 0 represents a circle, and state its centre and radius.

证明 x² + y² − 8x + 2y + 1 = 0 表示一个圆，并写出其圆心和半径。

Complete the square for x and y:

对 x 和 y 分别配方：

(x − 4)² − 16 + (y + 1)² − 1 + 1 = 0

(x − 4)² + (y + 1)² = 16

Since the right-hand side is positive, the equation represents a real circle. The centre is (4, −1) and the radius is 4.

由于右边为正数，该方程表示一个真实的圆。圆心为 (4, −1)，半径为 4。

---

7. Common Exam Pitfalls | 常见考试陷阱

Students often make the following mistakes when working with circle equations:

学生在处理圆的方程时经常犯以下错误：

• Forgetting to take half of D and E when finding the centre from the general form. The centre is (−D/2, −E/2), not (−D, −E).
• 忘记在用一般方程求圆心时需要取 D 和 E 的一半。圆心为 (−D/2, −E/2)，而不是 (−D, −E)。
• Mixing up the signs when reading the centre from the standard form. In (x − a)² + (y − b)², the centre is (a, b), so (x + 3)² means a = −3.
• 在从标准方程读圆心时搞错符号。在 (x − a)² + (y − b)² 中，圆心为 (a, b)，因此 (x + 3)² 表示 a = −3。
• Forgetting to take the square root of the right-hand side when finding the radius from the standard form.
• 在从标准方程求半径时忘记对右边开平方根。
• Dividing the entire general equation by a common factor but forgetting to divide the constant term as well.
• 将整个一般方程除以公因数时，忘记同时除以常数项。
• Not checking whether the equation actually represents a real circle before attempting to find the centre and radius.
• 在求圆心和半径之前，没有检验方程是否确实表示一个真实的圆。

---

8. Practice Questions | 巩固练习

Try these problems yourself before checking the answers.

请先自己尝试以下练习，再核对答案。

• Question 1: Convert (x + 5)² + (y − 2)² = 49 to general form.
• 第 1 题：将 (x + 5)² + (y − 2)² = 49 化为一般方程。
• Question 2: Convert x² + y² − 10x + 4y − 20 = 0 to standard form.
• 第 2 题：将 x² + y² − 10x + 4y − 20 = 0 化为标准方程。
• Question 3: Determine whether x² + y² + 2x − 6y + 15 = 0 represents a real circle.
• 第 3 题：判断 x² + y² + 2x − 6y + 15 = 0 是否表示一个真实的圆。

Answers | 答案

Answer 1: Expanding gives x² + 10x + 25 + y² − 4y + 4 = 49, so x² + y² + 10x − 4y − 20 = 0.

第 1 题答案：展开得 x² + 10x + 25 + y² − 4y + 4 = 49，即 x² + y² + 10x − 4y − 20 = 0。

Answer 2: Completing the square gives (x − 5)² + (y + 2)² = 49, so the centre is (5, −2) and r = 7.

第 2 题答案：配方得 (x − 5)² + (y + 2)² = 49，即圆心为 (5, −2)，r = 7。

Answer 3: Completing the square gives (x + 1)² + (y − 3)² = −5. Since the right-hand side is negative, the equation does not represent a real circle.

第 3 题答案：配方得 (x + 1)² + (y − 3)² = −5。由于右边为负数，该方程不表示真实的圆。

---

9. Summary | 总结

Mastering the conversion between the standard and general forms of a circle’s equation is essential for success in coordinate geometry. Remember the following key points:

掌握圆的标准方程与一般方程之间的转化，是在坐标几何中取得成功的关键。请记住以下要点：

Form | 形式	Equation | 方程	Centre | 圆心	Radius | 半径
Standard | 标准	(x − a)² + (y − b)² = r²	(a, b)	r
General | 一般	x² + y² + Dx + Ey + F = 0	(−D/2, −E/2)	√((D² + E²)/4 − F)

To convert from standard to general, expand and rearrange. To convert from general to standard, complete the square for x and y. Always check the sign of r² to ensure the equation represents a real circle.

从标准方程到一般方程，展开并整理即可；从一般方程到标准方程，需要对 x 和 y 配方。始终检查 r² 的符号，以确保方程表示真实的圆。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）