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1. Analytical Fundamentals

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Midpoint = [(x₁ + x₂)/2, (y₁ + y₂)/2]
Slope (m) = (y₂ - y₁) / (x₂ - x₁)
Area of Δ = 0.5 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Internal Division = [(mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)]
External Division = [(mx₂ - nx₁)/(m-n), (my₂ - ny₁)/(m-n)]

1.5 Angle Between Two Lines

For two lines with slopes m₁ and m₂, the angle θ between them is:

tanθ = | (m₁ - m₂) / (1 + m₁m₂) |

Special Conditions:

Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = -1

2. Standard Forms of a Line

General Form: ax + by + c = 0
Slope-Intercept: y = mx + c
Slope-Point Form: y - y₁ = m(x - x₁)
Two-Point Form: (y - y₁) = [(y₂ - y₁)/(x₂ - x₁)] * (x - x₁)
Double-Intercept: x/a + y/b = 1
Perpendicular: xcosα + ysinα = p

3. Pair of Lines, Circles & Conics

Pair of Straight Lines:

ax² + 2hxy + by² + 2gx + 2fy + c = 0

Circle (Standard Form):

(x - h)² + (y - k)² = r²

Parabola:

y = ax² + bx + c

Hyperbola:

y = a/x + q