Analytic Geometry Quick Reference

Coordinate Geometry

• Distance: d = sqrt((x2-x1)^2 + (y2-y1)^2)
• Midpoint: M = ((x1+x2)/2, (y1+y2)/2)
• Slope: m = (y2-y1)/(x2-x1)

Lines

• Slope-intercept: y = mx + b
• Point-slope: y - y1 = m(x - x1)
• Standard form: Ax + By = C
• Parallel lines have equal slopes.
• Perpendicular nonvertical lines satisfy m1*m2 = -1.

Conic Sections

• Circle: (x-h)^2 + (y-k)^2 = r^2
• Parabola: (x-h)^2 = 4p(y-k) or (y-k)^2 = 4p(x-h)
• Ellipse: (x-h)^2/a^2 + (y-k)^2/b^2 = 1
• Hyperbola: (x-h)^2/a^2 - (y-k)^2/b^2 = 1

Vectors

• Magnitude: |v| = sqrt(vx^2 + vy^2)
• Dot product: u·v = uxvx + uyvy
• Perpendicular vectors satisfy u·v = 0.

Polar Coordinates

• x = r cos(theta)
• y = r sin(theta)
• r^2 = x^2 + y^2

Three-Dimensional Geometry

• Distance: d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
• Plane: Ax + By + Cz = D

Use the individual chapters for explanations, derivations, examples, exercises, and applications.