Parametrizing Conic Sections

Every conic equation is really a disguised Pythagorean-style identity, and parametrizing it just means finding two functions of t that automatically satisfy that identity. For an ellipse (x-h)²/a² + (y-k)²/b² = 1, the left side is a sum of two squares equal to 1. Cosine and sine were built for exactly this: cos²(t) + sin²(t) = 1, always, for any t. So set (x-h)/a = cos(t) and (y-k)/b = sin(t), solve for x and y, and the equation is satisfied for every value of t. As t runs from 0 to 2pi, the point (x(t), y(t)) sweeps the whole ellipse once, centered at (h, k) and stretched by a horizontally and b vertically.

A hyperbola x²/a² - y²/b² = 1 has a difference of squares equal to 1, not a sum. That rules out sine and cosine but matches a different identity: sec²(t) - tan²(t) = 1. Setting x/a = sec(t) and y/b = tan(t) makes the equation true for every t where secant is defined. Because secant has a range of (-infinity,-1] union [1, infinity), x(t) = a sec(t) never lands between -a and a, which is exactly why the graph splits into two branches opening left and right instead of forming one closed loop.

This topic connects directly to the broader idea of parametrizing curves from Unit 4: once a curve's equation matches a known identity, writing x(t) and y(t) is almost automatic. It also links back to the unit circle and right-triangle trig ratios, since cos, sin, sec, and tan are the same functions doing the same job here as they do there, just scaled and shifted to fit a particular ellipse or hyperbola.