4.3 Parametrically Defined Circles and Lines

Parametric lines

A line through point (x₀, y₀) with direction (a, b) can be written as x(t) = x₀ + at, y(t) = y₀ + bt. As t varies, the point traces the line at constant rate. The ratio b/a gives the line's slope, and (a, b) is the direction vector.

Parametric circles

A circle of radius r centered at (h, k) is given by x(t) = h + r·cos(t), y(t) = k + r·sin(t). Since cos²t + sin²t = 1, eliminating t gives (x−h)² + (y−k)² = r². The parameter t represents the angle from the center, so the point moves counterclockwise as t increases.

Orientation and restricted domains

Parametric equations encode direction of motion, not just the graph's shape. Reversing signs (e.g., using −sin(t) or −t) reverses the direction of travel. Restricting the domain of t traces only part of the curve, such as a semicircle or a segment rather than a full line/circle.

Key terms

Parametric equations for a line through (x₀, y₀) with direction (a, b)

x(t) = x₀ + at, y(t) = y₀ + bt

Parametric equations for a circle centered at (h, k) with radius r

x(t) = h + r·cos(t), y(t) = k + r·sin(t)

How to eliminate the parameter t for a circle

Use cos²t + sin²t = 1 to derive (x-h)² + (y-k)² = r²

Effect of reversing sign of t (e.g., -t) in circle parametrization

Reverses the direction of motion (counterclockwise becomes clockwise)

Effect of restricting the domain of t

Traces only part of the curve, such as a line segment or arc instead of the full line/circle

Try an AP-style question

Which parametrizes the circle with center (2, −1) and radius 3?

1. x = 2cos 3t, y = −sin 3t
2. x = 2 + cos t, y = −1 + sin t
3. x = 2 + 3cos t, y = −1 + 3sin t
4. x = 3 + 2cos t, y = 3 − sin t