斜率xiélǜ

Core Concept

The slope (斜率) of a line measures how steeply it rises or falls. It is the single number that captures a line's direction in the coordinate plane, and it appears in almost every CSCA analytic-geometry question — either directly, or hidden inside a question about parallel lines, perpendicular lines, or tangent lines.

Mathematical Definition

Given two distinct points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ on a line, the slope $k$ is the ratio of vertical change to horizontal change:

$k = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2)$

This ratio is often read as "rise over run" (纵坐标之差比横坐标之差).

Slope and Inclination Angle

Every line has an inclination angle (倾斜角) $\alpha$, measured counterclockwise from the positive $x$-axis, with $0° \leq \alpha < 180°$. Slope and inclination angle are linked by:

$k = \tan\alpha$

This connection is why 倾斜角 questions and 斜率 questions are really the same question in two costumes:

Inclination angle $\alpha$	Slope $k$	Line direction
$\alpha = 0°$	$k = 0$	horizontal
$0° < \alpha < 90°$	$k > 0$	rising
$\alpha = 90°$	undefined	vertical
$90° < \alpha < 180°$	$k < 0$	falling

The vertical case is the trap. When $x_1 = x_2$ the denominator is zero, so the slope does not exist — the line is vertical, written $x = c$. A line with "no slope" is not a line with "slope $0$".

Important Properties

Property 1: Parallel Lines

Two non-vertical lines are parallel exactly when their slopes are equal:

$l_1 \parallel l_2 \iff k_1 = k_2 \ \text{(and different intercepts)}$

Property 2: Perpendicular Lines

Two lines with slopes are perpendicular exactly when the product of their slopes is $-1$:

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

Equivalently, $k_2 = -\dfrac{1}{k_1}$ — the negative reciprocal (负倒数). The exception: a vertical line ($x = c$) and a horizontal line ($y = c$) are perpendicular, but the product rule cannot be applied because one slope is undefined.

Property 3: Collinearity

Three points $A$, $B$, $C$ are collinear (共线) if and only if

$k_{AB} = k_{AC}$

This is the fastest way to test collinearity without vectors.

Worked Applications

Application 1: Road Gradient

A road sign reading "8% grade" means the road rises 8 metres for every 100 metres travelled horizontally — slope $k = 0.08$, inclination angle $\alpha = \arctan(0.08) \approx 4.6°$.

Application 2: Finding a Perpendicular Bisector

To find the perpendicular bisector of the segment joining $A(1, 3)$ and $B(5, 7)$:

1. Midpoint: $M\left(\frac{1+5}{2}, \frac{3+7}{2}\right) = (3, 5)$
2. Slope of $AB$: $k_{AB} = \frac{7-3}{5-1} = 1$
3. Slope of the bisector: $k = -\frac{1}{1} = -1$
4. Equation: $y - 5 = -1(x - 3)$, i.e. $x + y - 8 = 0$

CSCA Practice Problems

💡 Note: These problems follow the CSCA exam syllabus and Chinese standardized test formats.

Example 1: Basic (Difficulty ★★☆☆☆)

The line through $A(-1, 2)$ and $B(3, -6)$ has slope:

Options:

• A. $2$
• B. $-2$
• C. $\frac{1}{2}$
• D. $-\frac{1}{2}$

Detailed Solution:

Apply the slope formula directly:

$k = \frac{-6 - 2}{3 - (-1)} = \frac{-8}{4} = -2$

A common slip is to subtract in inconsistent orders, e.g. $\frac{-6-2}{-1-3} = 2$. Keep both differences in the same order: if $y_2$ comes first on top, $x_2$ must come first on the bottom.

Answer: B

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Example 2: Intermediate (Difficulty ★★★☆☆)

Line $l_1$ passes through $(0, 1)$ and $(2, m)$. Line $l_2$ has equation $2x + y - 3 = 0$. If $l_1 \perp l_2$, find $m$.

Options:

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Detailed Solution:

1. Rewrite $l_2$ in slope-intercept form: $y = -2x + 3$, so $k_2 = -2$.

2. Perpendicularity requires $k_1 \cdot k_2 = -1$: $k_1 \cdot (-2) = -1 \implies k_1 = \frac{1}{2}$

3. Compute $k_1$ from the two given points: $k_1 = \frac{m - 1}{2 - 0} = \frac{m-1}{2}$

4. Set equal and solve: $\frac{m-1}{2} = \frac{1}{2} \implies m - 1 = 1 \implies m = 2$

Answer: C

Common Mistakes

❌ Mistake 1: Saying a vertical line has slope 0

A vertical line has no slope (斜率不存在); a horizontal line has slope $0$. Exam questions deliberately include a vertical case to catch this. When a question says "line $l$ has slope $k$", it has already silently ruled out the vertical line — but when it says "line $l$ passes through point $P$", you must consider the vertical case separately.

❌ Mistake 2: Forgetting the parallel case when slopes are equal

$k_1 = k_2$ means the lines are parallel or identical. If the question asks for parallel lines, you must also check that the intercepts differ.

❌ Mistake 3: Confusing slope with inclination angle

Slope grows without bound as $\alpha \to 90°$, and the relationship $k = \tan\alpha$ is not linear. A line with $\alpha = 60°$ does not have "twice the slope" of a line with $\alpha = 30°$: $\tan 60° = \sqrt{3} \approx 1.73$, while $\tan 30° = \frac{\sqrt{3}}{3} \approx 0.58$.

Study Tips

1. ✅ Memorise the two-point formula — it is the foundation for every line problem
2. ✅ Always ask "could this line be vertical?" before dividing
3. ✅ Learn $k_1 k_2 = -1$ cold — perpendicularity is the most-tested slope property
4. ✅ Convert general form to slope-intercept form ($Ax + By + C = 0 \Rightarrow k = -\frac{A}{B}$, when $B \neq 0$) to read off the slope instantly
5. ✅ Sketch the line — a quick diagram catches sign errors immediately

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💡 Exam Tip: Slope appears in roughly a quarter of CSCA analytic-geometry questions, most often through the perpendicular condition $k_1 k_2 = -1$ and through tangent-line problems where the derivative supplies the slope.