Chapter in a nutshell: A straight line is described by its slope (gradient) $m$ and an intercept or a point. The main forms are $y=mx+c$, point–slope, and two-point. Parallel lines have equal slopes; perpendicular lines have slopes whose product is $-1$.
1. Key Ideas
- Slope/gradient $m$ = steepness = $\tan\theta$, where $\theta$ is the angle the line makes with the positive x-axis.
- Intercepts: where the line cuts the axes (y-intercept = c).
- A line is fixed by a point + a slope, or by two points.
2. Formulas (quick reference — all of them)
- Slope from two points: $m=\dfrac{y_2-y_1}{x_2-x_1}$
- Slope and angle: $m=\tan\theta$
- Slope–intercept form: $y=mx+c$ (c = y-intercept)
- Point–slope form: $y-y_1=m(x-x_1)$
- Two-point form: $\dfrac{y-y_1}{x-x_1}=\dfrac{y_2-y_1}{x_2-x_1}$
- General form: $ax+by+c=0$, with slope $=-\dfrac{a}{b}$
- Parallel lines: $m_1=m_2$
- Perpendicular lines: $m_1\times m_2=-1$
- Special lines: x-axis $y=0$; y-axis $x=0$; horizontal line $y=k$ (slope 0); vertical line $x=k$ (slope undefined).
- (Prerequisite) Mid-point/section, distance formulas (often combined in problems).
3. Prerequisite Ideas
- Cartesian coordinates; section/mid-point formula; basic trigonometry ($\tan\theta$).
4. Worked Example (one, for the method)
Q. Find the equation of the line through (1, 2) and (3, 8). Solution: $m=\dfrac{8-2}{3-1}=3$. Using point–slope: $y-2=3(x-1)\Rightarrow y=3x-1$.5. Common Mistakes to Avoid
- Subtracting coordinates in different orders in the slope (keep $y$ over $x$ consistently).
- For perpendiculars, using $m_1=m_2$ instead of $m_1m_2=-1$.
- Forgetting a vertical line has an undefined slope (equation $x=k$).
- Sign error when finding slope $-a/b$ from $ax+by+c=0$.
6. Likely Exam Questions (with crisp answers)
- Define slope. → Steepness, $m=\tan\theta=\frac{y_2-y_1}{x_2-x_1}$.
- Write the slope–intercept form. → $y=mx+c$.
- Write the point–slope form. → $y-y_1=m(x-x_1)$.
- Condition for two lines to be parallel? → Equal slopes ($m_1=m_2$).
- Condition for perpendicular lines? → $m_1m_2=-1$.
- Slope of the x-axis? → 0.
- Slope of a vertical line? → Undefined.
- Slope of $ax+by+c=0$? → $-a/b$.
- Equation of a line through (0,3) with slope 2? → $y=2x+3$.
- What does c represent in $y=mx+c$? → The y-intercept.
Extended & Prerequisite Formulas (full)
- Slope $m=\tfrac{y_2-y_1}{x_2-x_1}=\tan\theta$; forms: $y=mx+c$; $y-y_1=m(x-x_1)$; two-point $\tfrac{y-y_1}{x-x_1}=\tfrac{y_2-y_1}{x_2-x_1}$; general $ax+by+c=0$ (slope −a/b).
- Parallel: m₁=m₂; perpendicular: m₁m₂=−1; x-axis y=0; y-axis x=0; horizontal y=k; vertical x=k.
- Prereq: distance & section formulas; $\tan$ of standard angles (0,30,45,60,90).