Analytic geometry

The topic is divided into the following subtopics:
  • Points and vectors
  • Analytic geometry in a plane
  • Analytic geometry in a space
  • Conics

Points and vectors

Part I:
  • Points and vectors in plane and in space
  • Length of a vector
  • Operations with vectors – sum, scalar multiple
  • Linear combination of vectors
  • Linear dependence of vectors
  • Line segment – center, length
  • Triangle – centroid, centers of sides, lengths of sides, perimeter
Part II:
  • Scalar product (dot product) of vectors in plane and in space
  • Perpendicular vectors
  • Angle of vectors
  • Applications – plane shapes, solids in coordinate system
Part III:
  • Vector product of vectors
  • Area of a plane region, area of a face of a solid
  • Volume of a solid (parallelepiped, pyramid, tetrahedron)
  • Complex problems covering whole topic 

Analytic geometry in a plane

Part I:
  • Line – parametric description, general equation, point-slope form equation
  • Direction vector and normal vector of a line
  • Line segment, half-line – parametric description
  • Relative position of two lines
  • Perpendicularity of lines
  • Parallelity of lines
Part II:
  • Distance of a point from a line
  • Distance of two parallel lines
  • Angle of two lines
  • Triangle – medians, heights (altitudes), side perpendicular bisectors
  • Line and point reflection, translation
Part III:
  • Angles and distances – more complex problems
  • Complex problems covering whole topic

Analytic geometry in a space

Part I:
  • Line – parametric description
  • Plane - parametric description, general equation
  • Intersection of two lines
  • Intersection of a line and a plane
  • Intersection of two planes
  • Relative position of points, lines and planes
Part II:
  • Intersection of two planes – more complex problems
  • Perpendicularity of lines and planes
  • Parallelity of lines and planes
  • Angles of lines and planes
Part III:
  • Complex problems on perpendicularity
  • Point, line and plane reflection
  • Distance of a point from a plane
  • Distance of a point from a line
  • Metric problems on solids


Part I:
  • Circle (center and radius)
  • Ellipse (center, semi-major and semi-minor axis, foci, vertex and co-vertex)
Part II:
  • Parabola (vertex, directrix, focus)
  • Hyperbola (center, foci, vertices, semi-major and semi-minor axis, eccentricity)
Part III:
  • Tangent line to a conic
  • Conic and a line
  • Conic passing through given points