Unit 8 of 10

Conics, Parametric Equations, and Space

  • 9 lessons
  • Parts 183 to 200

Parabolas, ellipses and hyperbolas can be defined by distances to fixed points and lines, and each one has a standard equation. This unit builds those equations, shifts conics by completing the square, removes the mixed term by rotating the axes, and writes conics in polar form.

Parametric equations describe a path by giving both coordinates as functions of a parameter, which records the direction of travel as well as the shape. The unit shows how to sketch such paths and eliminate the parameter, and then steps into three dimensions with coordinates, distance and spheres.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Conic Sections

  1. Parabolas as Geometric Loci

    How a parabola is defined as the set of points equally far from a focus and a directrix, and how to find its vertex, focus, directrix and latus rectum from an equation.

    Theory part 183Practice part 184

  2. Ellipses as Geometric Loci

    How an ellipse is the set of points whose distances to two foci add up to a constant, and how its center, vertices, axes and eccentricity all follow from that one rule.

    Theory part 185Practice part 186

  3. Hyperbolas as Geometric Loci

    How a hyperbola is the set of points whose distances to two foci differ by a constant, and how its vertices, axes, asymptotes and eccentricity come from the three numbers a, b and c.

    Theory part 187Practice part 188

  4. Shifted Conics and Completing the Square

    How moving a conic changes its equation, how completing the square turns an expanded equation back into a shifted standard form, and how vertices, foci and asymptotes move with the curve.

    Theory part 189Practice part 190

  5. Rotation of Axes

    How turning the coordinate axes changes the coordinates of points, why a tilted conic has an xy term, how to choose the angle that removes it, and what the discriminant can and cannot tell you.

    Theory part 191Practice part 192

  6. Polar Equations of Conics

    How the focus and directrix rule becomes one polar equation when the focus sits at the pole, and how to read the eccentricity, the directrix and the vertices from that equation.

    Theory part 193Practice part 194

Parametric Equations

  1. Parametric Equations and Oriented Paths

    How two functions of one parameter describe a path in the plane, with a start, an end, a direction of travel and a pace, and why one curve can be traced in many different ways.

    Theory part 195Practice part 196

  2. Eliminating the Parameter

    How to remove the parameter from a pair of parametric equations to get an equation in x and y, and how to keep track of the restrictions, branches and direction that the new equation forgets.

    Theory part 197Practice part 198

Three Dimensions

  1. 3D Coordinate Systems and Distance Formula

    How three perpendicular axes give every point in space an ordered triple, how the coordinate planes split space into octants, and how to find distances, midpoints and the equation of a sphere.

    Theory part 199Practice part 200