Straight Lines
Formula Sheet 8
1. Basic Formulas
| Concept | Formula |
|---|---|
| Distance Formula (Between $$ P(x_1, y_1) $$ & $$ Q(x_2, y_2) $$) |
$$ PQ = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} $$ |
| Section Formula (Internal Division m:n) |
$$ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $$ |
| Area of Triangle | $$ \frac{1}{2} | x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2) | $$ |
2. Slope & Angles
Slope (m):
- If angle $$ \theta $$ given: $$ m = \tan \theta $$
- Two points given: $$ m = \frac{y_2 – y_1}{x_2 – x_1} $$
Angle Between Two Lines:
$$ \tan \theta = \left| \frac{m_1 – m_2}{1 + m_1 m_2} \right| $$
• Parallel: $$ m_1 = m_2 $$
• Perpendicular: $$ m_1 m_2 = -1 $$
3. Standard Equations
| Form Name | Equation |
|---|---|
| General Form | $$ ax + by + c = 0 $$ |
| Slope-Intercept | $$ y = mx + c $$ (c is y-intercept) |
| Point-Slope | $$ y – y_1 = m(x – x_1) $$ |
| Two-Point Form | $$ y – y_1 = \frac{y_2 – y_1}{x_2 – x_1}(x – x_1) $$ |
| Intercept Form | $$ \frac{x}{a} + \frac{y}{b} = 1 $$ |
| Normal Form | $$ x \cos \alpha + y \sin \alpha = p $$ |
Special Cases:
• X-Axis: $$ y = 0 $$ | Y-Axis: $$ x = 0 $$
• Parallel to X-Axis: $$ y = b $$ | Parallel to Y-Axis: $$ x = a $$
• X-Axis: $$ y = 0 $$ | Y-Axis: $$ x = 0 $$
• Parallel to X-Axis: $$ y = b $$ | Parallel to Y-Axis: $$ x = a $$
4. Advanced Concepts
Distance: Point to Line
Distance of $$ (x_1, y_1) $$ from $$ ax+by+c=0 $$:
$$ d = \left| \frac{ax_1 + by_1 + c}{\sqrt{a^2 + b^2}} \right| $$
Distance: Parallel Lines
Between $$ ax+by+c_1=0 $$ & $$ ax+by+c_2=0 $$:
$$ d = \left| \frac{c_1 – c_2}{\sqrt{a^2 + b^2}} \right| $$
Shifting of Origin:
If origin is shifted to $$ (h, k) $$, the new coordinates $$ (X, Y) $$ are related to old $$ (x, y) $$ by:
$$ x = X + h \quad \text{and} \quad y = Y + k $$