Intersections of Straight Lines and Circles | 直线与圆的交点

📚 Intersections of Straight Lines and Circles | 直线与圆的交点

In A-Level mathematics, finding the intersection points between a straight line and a circle is a fundamental skill that combines algebra and geometry. You will often need to solve a linear equation and a quadratic equation simultaneously, then interpret the discriminant to decide how many intersection points exist.

在 A-Level 数学中，求直线与圆的交点是代数与几何结合的基本技能。通常需要联立一个一次方程和一个二次方程，并通过判别式判断交点个数。

1. Circle Equations: Standard and General Form | 圆的方程：标准式与一般式

The standard equation of a circle with centre \((a,b)\) and radius \(r\) is written as \((x-a)^2 + (y-b)^2 = r^2\). The general form expands this to \(x^2 + y^2 + Dx + Ey + F = 0\), where \(D = -2a\), \(E = -2b\), and \(F = a^2 + b^2 – r^2\).

圆心为 \((a,b)\)、半径为 \(r\) 的圆的标准方程为 \((x-a)^2 + (y-b)^2 = r^2\)。一般式展开为 \(x^2 + y^2 + Dx + Ey + F = 0\)，其中 \(D = -2a\)，\(E = -2b\)，\(F = a^2 + b^2 – r^2\)。

When solving intersection problems, you may be given either form. It is often easiest to keep the circle in standard form because substitution becomes cleaner.

在求解交点问题时，题目可能给出任意一种形式。通常保留标准式更便于代入化简。

2. Line Equations: Slope-Intercept and General Form | 直线方程：斜截式与一般式

A straight line can be written as \(y = mx + c\) (slope-intercept form) or as \(ax + by + c = 0\) (general form). For intersection problems, the slope-intercept form is convenient because you can substitute \(y\) directly into the circle equation.

直线可写作 \(y = mx + c\)（斜截式）或 \(ax + by + c = 0\)（一般式）。在求交点时，斜截式更为方便，因为可直接将 \(y\) 代入圆方程。

If the line is vertical, \(x = k\), substitution is even simpler: replace \(x\) with \(k\) in the circle equation and solve a quadratic in \(y\).

若直线为竖直线 \(x = k\)，代入更加简单：将圆方程中的 \(x\) 换成 \(k\)，然后解关于 \(y\) 的二次方程。

3. Substitution Method: Setting Up the Equation | 代入法：建立方程

To find intersections, substitute the expression for \(y\) from the line equation into the circle equation. For example, if the line is \(y = mx + c\), replace every \(y\) in the circle equation by \(mx + c\). This produces a quadratic equation in \(x\).

求交点时，将直线方程中的 \(y\) 表达式代入圆方程。例如直线为 \(y = mx + c\)，将圆方程中所有 \(y\) 替换为 \(mx + c\)，得到关于 \(x\) 的二次方程。

After expanding and collecting like terms, the quadratic takes the form \(Ax^2 + Bx + C = 0\). Solving this quadratic gives the \(x\)-coordinates of the intersection points.

展开并合并同类项后，二次方程化为 \(Ax^2 + Bx + C = 0\)。解该方程即可得到交点的 \(x\) 坐标。

4. The Discriminant: Key to the Number of Intersections | 判别式：交点个数的关键

For the quadratic \(Ax^2 + Bx + C = 0\), the discriminant is \(\Delta = B^2 – 4AC\). The value of \(\Delta\) tells us exactly how many intersection points exist between the line and the circle.

对于二次方程 \(Ax^2 + Bx + C = 0\)，判别式为 \(\Delta = B^2 – 4AC\)。\(\Delta\) 的值直接告诉我们直线与圆交点的个数。

\(\Delta > 0\): two distinct real roots, so the line cuts the circle at two points (a secant).
\(\Delta = 0\): one repeated root, so the line touches the circle at exactly one point (a tangent).
\(\Delta < 0\): no real roots, so the line does not meet the circle at all.

\(\Delta > 0\)：两个不等实根，直线与圆相交于两点（割线）。
\(\Delta = 0\)：一个重根，直线与圆相切于一点（切线）。
\(\Delta < 0\)：无实根，直线与圆没有交点。

5. Geometric Interpretation: Distance from Centre | 几何法：圆心到直线的距离

Alternatively, you can determine the number of intersections by comparing the perpendicular distance \(d\) from the centre of the circle to the line with the radius \(r\). The line is given by \(ax + by + c = 0\), and the distance formula is:

另一种方法是比较圆心到直线的垂直距离 \(d\) 与半径 \(r\)。设直线为 \(ax + by + c = 0\)，距离公式为：

\(d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}\)

Here \((x_0, y_0)\) is the centre of the circle. If \(d < r\), the line is a secant; if \(d = r\), the line is a tangent; if \(d > r\), the line misses the circle.

其中 \((x_0, y_0)\) 是圆心坐标。若 \(d < r\)，直线为割线；若 \(d = r\)，直线为切线；若 \(d > r\)，直线与圆相离。

This method is often faster than substitution, especially when you only need the number of intersections and not their coordinates.

此方法通常比代入法更快，尤其当只需判断交点个数而不需求坐标时。

6. Tangent Case: One Point of Contact | 相切情形：一个切点

When the discriminant is zero, the line is tangent to the circle. The single intersection point is the point of tangency. To find its coordinates, you solve the quadratic using \(x = -B/(2A)\), then substitute back to find \(y\).

当判别式为零时，直线与圆相切。唯一交点即为切点。求坐标时，利用 \(x = -B/(2A)\) 解二次方程，再代回直线求 \(y\)。

A useful geometric fact: at the point of tangency, the radius to that point is perpendicular to the tangent line. The gradient of the radius is the negative reciprocal of the gradient of the tangent.

一个常用的几何性质：在切点处，过切点的半径与切线垂直，半径的斜率是切线斜率的负倒数。

7. Secant Case: Two Distinct Points | 割线情形：两个不同交点

When \(\Delta > 0\), the line intersects the circle in two points. The quadratic formula gives two \(x\)-values: \(x = \frac{-B \pm \sqrt{\Delta}}{2A}\). Substitute each into the line equation to obtain the corresponding \(y\)-values.

当 \(\Delta > 0\) 时，直线与圆相交于两点。二次公式给出两个 \(x\) 值：\(x = \frac{-B \pm \sqrt{\Delta}}{2A}\)。将每个 \(x\) 代入直线方程得到相应的 \(y\) 值。

If the line is vertical, \(x = k\), substitute \(k\) into the circle equation to get a quadratic in \(y\). The two roots give the two \(y\)-coordinates directly.

若直线为竖直线 \(x = k\)，将 \(k\) 代入圆方程得到关于 \(y\) 的二次方程。两个根直接给出两个交点的 \(y\) 坐标。

8. No Intersection: \(\Delta < 0\) | 无交点：\(\Delta < 0\)

If the discriminant is negative, the quadratic has no real solutions, meaning the line does not touch or cross the circle. The line lies entirely outside the circle. You may be asked to show this algebraically or to find the range of \(m\) for which the line \(y = mx + c\) does not meet the circle.

若判别式为负，二次方程无实数解，说明直线与圆既不相交也不相切，直线完全在圆外。题目可能要求你代数地证明这一点，或求直线 \(y = mx + c\) 不与圆相交时斜率 \(m\) 的取值范围。

For example, set \(\Delta < 0\) and solve the resulting inequality for the unknown parameter.

例如，令 \(\Delta < 0\)，解关于未知参数的不等式即可。

9. Finding the Chord Length | 求弦长

When a line cuts a circle at two points, the segment between those points is called a chord. The length of the chord can be found using the distance formula between the two intersection points.

当直线与圆相交于两点时，连接两交点的线段称为弦。弦长可用两点间的距离公式计算。

Alternatively, if you know the centre distance \(d\) and the radius \(r\), the chord length is given by:

或者，若已知圆心到直线的距离 \(d\) 和半径 \(r\)，弦长公式为：

\(\text{Chord length} = 2\sqrt{r^2 – d^2}\)

This formula is derived from the right triangle formed by the radius, half the chord, and the perpendicular distance.

该公式由半径、半弦长和垂距组成的直角三角形推导而来。

10. Worked Example: Two Intersections | 典型例题：两个交点

Consider the circle \((x-1)^2 + (y+2)^2 = 25\) and the line \(y = 2x + 1\). Substitute \(y\) into the circle:

考虑圆 \((x-1)^2 + (y+2)^2 = 25\) 和直线 \(y = 2x + 1\)。将 \(y\) 代入圆方程：

\((x-1)^2 + (2x+3)^2 = 25\)

Expanding gives \(x^2 – 2x + 1 + 4x^2 + 12x + 9 = 25\), so \(5x^2 + 10x – 15 = 0\), which simplifies to \(x^2 + 2x – 3 = 0\).

展开得 \(x^2 – 2x + 1 + 4x^2 + 12x + 9 = 25\)，即 \(5x^2 + 10x – 15 = 0\)，化简为 \(x^2 + 2x – 3 = 0\)。

Factorising: \((x+3)(x-1) = 0\), so \(x = -3\) or \(x = 1\). Substituting into \(y = 2x + 1\) gives \(y = -5\) and \(y = 3\). The intersection points are \((-3, -5)\) and \((1, 3)\).

因式分解得 \((x+3)(x-1) = 0\)，所以 \(x = -3\) 或 \(x = 1\)。代入 \(y = 2x + 1\) 得 \(y = -5\) 和 \(y = 3\)。交点坐标为 \((-3, -5)\) 和 \((1, 3)\)。

11. Worked Example: Tangent Condition | 典型例题：相切条件

Find the value(s) of \(c\) such that the line \(y = 3x + c\) is tangent to the circle \(x^2 + y^2 = 10\).

求常数 \(c\) 的值，使直线 \(y = 3x + c\) 与圆 \(x^2 + y^2 = 10\) 相切。

Substitute \(y = 3x + c\) into \(x^2 + y^2 = 10\):

将 \(y = 3x + c\) 代入 \(x^2 + y^2 = 10\)：

\(x^2 + (3x + c)^2 = 10 \Rightarrow 10x^2 + 6cx + c^2 – 10 = 0\)

For tangency, the discriminant must be zero: \(\Delta = (6c)^2 – 4(10)(c^2 – 10) = 0\).

相切要求判别式为零：\(\Delta = (6c)^2 – 4(10)(c^2 – 10) = 0\)。

This simplifies to \(36c^2 – 40c^2 + 400 = 0\), so \(-4c^2 + 400 = 0\), giving \(c = \pm 10\).

化简得 \(36c^2 – 40c^2 + 400 = 0\)，即 \(-4c^2 + 400 = 0\)，解得 \(c = \pm 10\)。

12. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧

One common mistake is forgetting to expand the squared term correctly when the line has a fractional gradient. Always write out the substitution and simplify step by step.

常见错误之一是在直线斜率为分数时没有正确展开平方项。务必写出代换过程并逐步化简。

Another pitfall is ignoring the \(\pm\) sign when taking square roots in the geometric distance method. Remember that distance is always non-negative.

另一个陷阱是在几何距离法中求平方根时忽略 \(\pm\) 符号。记住距离总是非负的。

In exams, check whether the question asks for exact values or decimal approximations. Also, always state whether the line is a secant, tangent, or non-intersecting, and justify it with the discriminant or distance comparison.

考试中，注意题目要求精确值还是近似值。同时，务必说明直线是割线、切线还是相离，并用判别式或距离比较加以证明。

Finally, when using the distance method, ensure the line equation is written in the form \(ax + by + c = 0\) before applying the formula; otherwise, the sign and denominator will be wrong.

最后，使用距离法时，确保直线方程写成 \(ax + by + c = 0\) 的形式，否则符号和分母都会出错。

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Intersections of Straight Lines and Circles | 直线与圆的交点