Mathematics · Ch 9 — Straight Lines

Summary

Slope of a line: $m = \tan\theta$ where $\theta$ is the inclination ( $0 \le \theta < \pi$ ). For two points $(x_1, y_1)$ and $(x_2, y_2)$ , $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Parallel lines: $m_1 = m_2$ . Perpendicular lines: $m_1 m_2 = -1$ .
Angle between two lines: $\tan\theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$ .
Equation forms:
- Slope-intercept: $y = mx + c$
- Point-slope: $y - y_1 = m(x - x_1)$
- Two-point: $\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}$
- Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$
- Normal form: $x\cos\alpha + y\sin\alpha = p$
- General form: $Ax + By + C = 0$
Distance from a point $(x_1, y_1)$ to line $Ax + By + C = 0$ : $\frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ .
Distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ : $\frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$ . …