Parametrically Defined Circles and Lines

Unit 4 · Functions Involving Parameters, Vectors, and Matrices

What AP Precalc asks here

Two basic parametrizations dominate this topic. A line through $(x_0, y_0)$ with direction $(a, b)$ has $x = x_0 + at, y = y_0 + bt$ . A circle of radius $r$ centered at $(h, k)$ has $x = h + r \cos t, y = k + r \sin t$ for $t \in [0, 2\pi)$ . Verifying a parametrization usually means eliminating $t$ and confirming the standard rectangular equation.

Parametrizations

Line through

(x_0, y_0)

with direction

(a, b)

x = x_0 + a t, \;\; y = y_0 + b t

Circle radius

r

, center

(h, k)

x = h + r \cos t, \;\; y = k + r \sin t

AP Tip: For a circle, the trig functions can be swapped or sign-flipped — the curve is still the same circle, just traced in a different direction or starting position.

Type 1

Parametric line

Write $x_0 + at, y_0 + bt$ using the given point and direction.

Example 1

Parametrize the line through

(2, 3)

with direction

(1, -2)

. (A)

x = 2 + t, y = 3 - 2t

(B)

x = 2t + 1, y = 3t - 2

(C)

x = 1 + 2t, y = -2 + 3t

(D)

x = 2t, y = 3t - 2

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 2

Parametric circle

Set $x = h + r \cos t, y = k + r \sin t$ using the given center and radius.

Example 2

Parametrize the circle of radius 4 centered at

(1, -2)

. (A)

x = 4 \cos t, y = -2 + 4 \sin t

(B)

x = 1 + 4 \cos t, y = -2 + 4 \sin t

(C)

x = 4 + \cos t, y = 4 - 2 \sin t

(D)

x = 1 + \cos t, y = -2 + \sin t

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 3

Convert to rectangular

Eliminate $t$ using the Pythagorean identity (for a circle) or by solving (for a line).

Example 3

Convert

x = 1 + 2 \cos t, y = -1 + 2 \sin t

to rectangular. (A)

(x - 1)^2 + (y + 1)^2 = 4

(B)

(x + 1)^2 + (y - 1)^2 = 4

(C)

(x - 1)^2 + (y + 1)^2 = 2

(D)

x^2 + y^2 = 4

Practice more of this type— AI-generated · always-new problems

Generate Problems →