1. Distance & Section Formulas (Internal, External & Midpoint)
Distance between points A(x1, y1) and B(x2, y2): d = √[(x2 - x1)² + (y2 - y1)²]. Section Formula: The point P(x, y) dividing AB in ratio m:n: Internal division: x = (m·x2 + n·x1)/(m + n), y = (m·y2 + n·y1)/(m + n). External division: x = (m·x2 - n·x1)/(m - n), y = (m·y2 - n·y1)/(m - n). Midpoint: M = ((x1 + x2)/2, (y1 + y2)/2). Centroid of triangle with vertices A, B, C: G = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3).
2. Area of Triangle & Condition of Collinearity
Area of triangle formed by vertices (x1, y1), (x2, y2), and (x3, y3): Area = (1/2) |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. Condition of Collinearity: Three points lie on the same straight line if and only if the area of the triangle formed by them equals zero: x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) = 0.
3. Slope of a Line & Angle Between Two Lines
Slope (gradient m) of a line passing through (x1, y1) and (x2, y2): m = tan θ = (y2 - y1)/(x2 - x1) (where x1 ≠ x2). For general line Ax + By + C = 0, slope m = -A/B. Angle θ between two lines of slopes m1 and m2: tan θ = |(m2 - m1)/(1 + m1·m2)|. Parallel lines condition: m1 = m2. Perpendicular lines condition: m1 · m2 = -1 (or A1·A2 + B1·B2 = 0).
4. Standard Forms of Straight Line Equations
Fundamental line representations: (1) Slope-Intercept Form: y = mx + c (where m is slope, c is y-intercept); (2) Point-Slope Form: y - y1 = m(x - x1); (3) Two-Point Form: (y - y1)/(y2 - y1) = (x - x1)/(x2 - x1); (4) Intercept Form: x/a + y/b = 1 (where a and b are x- and y-intercepts); (5) Normal (Perpendicular) Form: x cos α + y sin α = p (where p is perpendicular distance from origin, α is angle of normal).
5. Perpendicular Distance of a Point from a Line
Distance from point P(x1, y1) to line Ax + By + C = 0: d = |A·x1 + B·y1 + C| / √(A² + B²). Distance between two parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0: d = |C1 - C2| / √(A² + B²). Notice that coefficients of x and y must be identical before subtracting constant terms.
6. Conic Sections Quick Reference (Circle & Parabola)
Standard Circle: Center (h, k) and radius r: (x - h)² + (y - k)² = r²; General form x² + y² + 2gx + 2fy + c = 0 has center (-g, -f) and radius r = √(g² + f² - c). Standard Parabola (vertex at origin): y² = 4ax (opens right, focus at (a, 0), directrix x = -a, latus rectum length = 4a); x² = 4ay (opens upwards, focus at (0, a)). Ellipse (x²/a² + y²/b² = 1, a > b): eccentricity e = √(1 - b²/a²), foci at (±ae, 0).