12.5 General forms and conic models

Ideas in this lesson

Classify a conic

With no cross-product term, compare squared-coordinate coefficients; complete the square to check real nondegenerate points.

Translate a general form

Complete the square in each coordinate, then divide by the resulting positive constant when needed.

General ellipses

With unequal positive squared coefficients, complete both squares and normalize the right side to one.

Eccentricity

Eccentricity is focal distance divided by the corresponding major or transverse semiaxis; it distinguishes ellipses from hyperbolas.

Reflector models

A parabolic reflector’s focus lies on its axis. Substitute an observed rim point to recover its focal parameter.