9.1 Equations of Straight Lines and Positional Relationships

9.1 Equations of Straight Lines and Positional Relationships

Analytic geometry bridges algebra and geometry by translating geometric figures into algebraic equations. In the Gaokao Mathematics examination, the straight line serves as the fundamental linear building block. Mastering the analytical properties of straight lines—their inclinations, slopes, structural equation forms, positional relationships, distance metrics, and reflection properties—is essential for solving both standalone coordinate geometry items and complex comprehensive conic section problems.

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1. Inclination Angle and Slope of a Line

Definition of Inclination Angle

In a rectangular Cartesian coordinate system $(xOy)$, when a straight line $l$ intersects the $x$-axis, the angle $\alpha$ formed by rotating the positive $x$-axis counterclockwise around the intersection point to coincide with the line $l$ is called the inclination angle (倾斜角) of the line $l$.

• By convention, when a line $l$ is parallel to or coincides with the $x$-axis, its inclination angle is defined as $\alpha = 0^\circ$ ($0$ radians).
• The range of the inclination angle $\alpha$ for any line in the coordinate plane is strictly restricted to: $$\alpha \in [0^\circ, 180^\circ) \quad \text{or} \quad \alpha \in [0, \pi)$$

Definition of Slope

When the inclination angle $\alpha \neq 90^\circ$ ($\alpha \neq \frac{\pi}{2}$), the tangent value of the inclination angle is defined as the slope (斜率) of the line $l$, denoted by $k$:

$$k = \tan \alpha$$

• When $\alpha \in [0^\circ, 90^\circ)$, $k \ge 0$, and $k$ increases monotonically from $0$ to $+\infty$ as $\alpha$ increases from $0^\circ$ toward $90^\circ$.
• When $\alpha = 90^\circ$, $\tan 90^\circ$ is undefined; hence vertical lines have no defined slope.
• When $\alpha \in (90^\circ, 180^\circ)$, $k < 0$, and $k$ increases monotonically from $-\infty$ to $0^-$ as $\alpha$ increases from $90^\circ$ toward $180^\circ$.

Two-Point Slope Formula

If a non-vertical line $l$ passes through two distinct points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ where $x_1 \neq x_2$, the slope $k$ is given by:

$$k = \frac{y_2 - y_1}{x_2 - x_1} = \frac{y_1 - y_2}{x_1 - x_2}$$

Proof: Let the directional vector of line $l$ be $\vec{v} = \vec{P_1 P_2} = (x_2 - x_1, y_2 - y_1)$. By definition of vector direction, $\tan \alpha = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$.

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2. The Five Standard Forms of Straight Line Equations

Depending on the given geometric parameters (such as points, slopes, or intercepts), a line can be expressed in five distinct algebraic forms:

Conversion to General Form

In Gaokao final submissions, final line equations should always be simplified into General Form $Ax + By + C = 0$ where $A, B, C$ are simplified coprime integers and $A \ge 0$.

• Slope of $Ax + By + C = 0$ (when $B \neq 0$): $k = -\frac{A}{B}$.
• $x$-intercept (when $A \neq 0$): $x = -\frac{C}{A}$; $y$-intercept (when $B \neq 0$): $y = -\frac{C}{B}$.
• Normal vector of the line: $\vec{n} = (A, B)$; Directional vector: $\vec{v} = (-B, A)$ or $(B, -A)$.

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3. Positional Relationships Between Two Lines

Let two lines be given by $l_1: A_1 x + B_1 y + C_1 = 0$ and $l_2: A_2 x + B_2 y + C_2 = 0$.

Parallelism Condition ($l_1 \parallel l_2$)

Two lines are parallel if and only if their directional vectors are collinear while they do not coincide:

$$l_1 \parallel l_2 \iff A_1 B_2 - A_2 B_1 = 0 \quad \text{and} \quad A_1 C_2 - A_2 C_1 \neq 0$$

If expressed in slope-intercept form $y = k_1 x + b_1$ and $y = k_2 x + b_2$:

$$l_1 \parallel l_2 \iff k_1 = k_2 \quad \text{and} \quad b_1 \neq b_2$$

Warning: If $A_1 B_2 - A_2 B_1 = 0$ and $A_1 C_2 - A_2 C_1 = 0$, the two lines are coincident (重合), not parallel.

Perpendicularity Condition ($l_1 \perp l_2$)

Two lines are perpendicular if and only if their normal vectors $\vec{n}_1 = (A_1, B_1)$ and $\vec{n}_2 = (A_2, B_2)$ are orthogonal ($\vec{n}_1 \cdot \vec{n}_2 = 0$):

$$l_1 \perp l_2 \iff A_1 A_2 + B_1 B_2 = 0$$

If both lines have defined slopes $k_1$ and $k_2$:

$$l_1 \perp l_2 \iff k_1 k_2 = -1$$

(Note: The general form $A_1 A_2 + B_1 B_2 = 0$ handles vertical/horizontal line pairs naturally without encountering division by zero.)

Inclination Angle Between Two Intersecting Lines

Let $\theta$ be the acute or right angle between $l_1$ and $l_2$ ($\theta \in (0^\circ, 90^\circ]$):

$$\cos \theta = \frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1| |\vec{n}_2|} = \frac{|A_1 A_2 + B_1 B_2|}{\sqrt{A_1^2 + B_1^2} \sqrt{A_2^2 + B_2^2}}$$

If slopes $k_1, k_2$ exist and $k_1 k_2 \neq -1$:

$$\tan \theta = \left| \frac{k_2 - k_1}{1 + k_1 k_2} \right|$$

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4. Distance Formulas and Distance Geometry

Point-to-Line Distance Formula

The perpendicular distance $d$ from a point $P(x_0, y_0)$ to a line $l: Ax + By + C = 0$ is:

$$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$

Rigorous Proof: Let $Q(x_1, y_1)$ be the projection of $P(x_0, y_0)$ onto $l$. The vector $\vec{QP}$ is parallel to the normal vector $\vec{n} = (A, B)$. Since $Q$ lies on $l$, $A x_1 + B y_1 + C = 0 \implies A x_1 + B y_1 = -C$. The scalar projection of $\vec{P_0 P}$ (where $P_0$ is any point on $l$) onto $\vec{n}$ gives:

$$d = \frac{|\vec{n} \cdot \vec{P_0 P}|}{|\vec{n}|} = \frac{|A(x_0 - x_1) + B(y_0 - y_1)|}{\sqrt{A^2 + B^2}} = \frac{|Ax_0 + By_0 - (Ax_1 + By_1)|}{\sqrt{A^2 + B^2}} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$

Distance Between Parallel Lines

For two parallel lines $l_1: Ax + By + C_1 = 0$ and $l_2: Ax + By + C_2 = 0$ ($C_1 \neq C_2$):

$$d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$$

Requirement: Both line equations must be normalized to share identical coefficients for $x$ and $y$ ($A$ and $B$) before applying this formula.

Family of Lines Passing Through an Intersection (Line Pencil)

Given two intersecting lines $l_1: A_1 x + B_1 y + C_1 = 0$ and $l_2: A_2 x + B_2 y + C_2 = 0$, the equation:

$$A_1 x + B_1 y + C_1 + \lambda (A_2 x + B_2 y + C_2) = 0 \quad (\lambda \in \mathbb{R})$$

represents the family of all straight lines passing through the intersection point of $l_1$ and $l_2$, excluding line $l_2$ itself.

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5. Reflection and Symmetry Problems in Cartesian Coordinates

Symmetry is a frequently tested topic in Gaokao analytic geometry. Two core scenarios exist:

Point Reflection Across a Line

To find the reflection point $P'(x', y')$ of $P(x_0, y_0)$ across mirror line $l: Ax + By + C = 0$:

1. Perpendicularity Condition: Slope of $PP'$ is perpendicular to line $l$, so $\vec{PP'} \parallel \vec{n} = (A, B)$, giving $\frac{y' - y_0}{x' - x_0} = \frac{B}{A}$ (or $(x' - x_0) B - (y' - y_0) A = 0$).
2. Midpoint Condition: The midpoint $M\left(\frac{x_0 + x'}{2}, \frac{y_0 + y'}{2}\right)$ of segment $PP'$ lies on $l$: $$A\left(\frac{x_0 + x'}{2}\right) + B\left(\frac{y_0 + y'}{2}\right) + C = 0$$

Solving this $2 \times 2$ system uniquely yields coordinates $(x', y')$.

Line Reflection Across a Line

To find the reflection of line $l_1$ across mirror line $l$:

• Find the intersection point $I = l_1 \cap l$ (if $l_1 \nparallel l$).
• Pick a non-intersection point $P \in l_1$, calculate its point reflection $P'$ across $l$.
• The reflected line is the unique line passing through $I$ and $P'$.