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How do you write parametric equations for a circle or a line, and how do the parameters control them?

Topic 4.4 Parametrically Defined Circles and Lines: write and interpret parametric equations for circles and lines, controlling radius, center, direction and starting point.

A focused answer to AP Precalculus Topic 4.4, covering the standard parametric forms for lines and circles, how radius, center, direction and starting point appear in the equations, and how to read or build them.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

A line through a point $(x_0, y_0)$ with direction $(a, b)$ is parametrised as $x(t) = x_0 + at$ , $y(t) = y_0 + bt$ : the start is the point, and the direction sets how $x$ and $y$ change per unit of $t$ . A circle of radius $r$ centered at the origin is $x(t) = r\cos t$ , $y(t) = r\sin t$ , because $x^2 + y^2 = r^2$ by the Pythagorean identity. Shifting the center to $(h, k)$ gives $x(t) = h + r\cos t$ , $y(t) = k + r\sin t$ . The parameter $t$ for a circle is the angle, sweeping counterclockwise as $t$ increases; swapping sine and cosine or negating the angle reverses the direction or changes the starting point. These two forms, the line and the circle, are the building blocks for the conic sections and motion models of the rest of Unit 4.

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What this topic is asking
Parametric lines
Parametric circles
Controlling direction and starting point
Try this

What this topic is asking

The College Board (Topic 4.4) wants you to write and interpret parametric equations for circles and lines. For a line, the parameter moves you from a starting point along a direction; for a circle, the parameter is an angle sweeping around the center. You control the radius, center, direction and starting point through the constants in the equations.

Parametric lines

The constants split cleanly: the constant terms give the starting point, and the coefficients of $t$ give the direction and speed. This is exactly the constant-rate motion model of Topic 4.2.

Parametric circles

So a circle is the unit circle scaled by $r$ and shifted by $(h, k)$ , with $t$ playing the role of the angle from Topic 3.2.

Building and checking a parametric circle

Write parametric equations for a circle of radius $4$ centered at $(-1, 3)$ , and verify the point at $t = \frac{\pi}{2}$ .

step 1 Apply the standard form

Center $(h, k) = (-1, 3)$ , radius $r = 4$ : $x(t) = -1 + 4\cos t$ , $y(t) = 3 + 4\sin t$ .

step 2 Evaluate at the quarter angle

At $t = \frac{\pi}{2}$ : $\cos\frac{\pi}{2} = 0$ , $\sin\frac{\pi}{2} = 1$ . So $x = -1 + 4(0) = -1$ and $y = 3 + 4(1) = 7$ .

step 3 Check it lies on the circle

Distance from the center: $\sqrt{(-1 - (-1))^2 + (7 - 3)^2} = \sqrt{0 + 16} = 4 = r$ . The point $(-1, 7)$ is exactly $4$ from the center, the topmost point.

step 4 Confirm the orientation

At $t = 0$ the point is $(3, 3)$ (rightmost); at $t = \frac{\pi}{2}$ it is $(-1, 7)$ (top). Moving from right to top is counterclockwise, as expected for increasing $t$ .

Controlling direction and starting point

A point worth stating once is how to change the direction and start of a parametric circle. Swapping to $x = h + r\sin t$ , $y = k + r\cos t$ starts at the top and sweeps clockwise; negating the angle, $x = h + r\cos t$ , $y = k - r\sin t$ , reverses the orientation to clockwise from the right. Scaling the parameter, as in $\cos(2t)$ , makes the point go around twice as fast (covering the circle in half the parameter range). Because the same circle can be parametrised in all these ways, always check both the starting point ( $t = 0$ ) and the direction (which way an early increase in $t$ moves the point) when matching equations to a described motion.

A second point worth keeping is how the line and circle forms differ in what the parameter does. For a line, the parameter enters linearly in both components, so equal steps in $t$ move equal distances along the line, at constant speed forever; the line never closes. For a circle, the parameter enters through sine and cosine, so the point moves around a closed loop and returns to its start every $2\pi$ ; equal steps in $t$ sweep equal angles, not equal arc lengths in any straight sense. This is why a line is parametrised by a direction vector while a circle is parametrised by an angle: the two forms answer different questions, "how far along the line" versus "how far around the loop". Recognizing which question a context asks tells you immediately whether to reach for the linear form or the trigonometric one.

Try this

Q1. Parametrise a circle of radius $1$ centered at $(0, 0)$ . [1 point]

Cue. $x(t) = \cos t$ , $y(t) = \sin t$ : the unit circle itself.

Q2. Write a line through $(0, 5)$ with direction $(2, -1)$ . [1 point]

Cue. $x(t) = 2t$ , $y(t) = 5 - t$ .

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2025 (style)1 marksSection I, Part A (multiple choice, no calculator). Which parametric equations trace a circle of radius

3

centered at the origin? (A)

x = 3\cos t,\ y = 3\sin t

(B)

x = 3t,\ y = 3t

(C)

x = 3 + \cos t,\ y = 3 + \sin t

(D)

x = \cos(3t),\ y = \sin(3t)

Show worked answer →

The correct answer is (A), $x = 3\cos t,\ y = 3\sin t$ .

For a circle of radius $r$ centered at the origin, the standard parametrisation is $x = r\cos t$ , $y = r\sin t$ , since then $x^2 + y^2 = r^2(\cos^2 t + \sin^2 t) = r^2$ . Here $r = 3$ . Choice (C) is a circle of radius $1$ centered at $(3, 3)$ , and (D) is the unit circle traced three times as fast.

AP 2025 (style)4 marksSection II (free response, no calculator). (a) Write parametric equations for the line through

(1, 2)

in the direction that increases

x

3

and

y

4

per unit of

t

. (b) Write parametric equations for a circle of radius

5

centered at

(2, -1)

Show worked answer →

A 4-point question on building parametric lines and circles.

(a) Line (2 points): start at $(1, 2)$ and add the direction times $t$ : $x(t) = 1 + 3t$ , $y(t) = 2 + 4t$ .
(b) Circle (2 points): a circle of radius $5$ about center $(2, -1)$ is $x(t) = 2 + 5\cos t$ , $y(t) = -1 + 5\sin t$ , since the center shifts the standard circle and the radius scales it.

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Sources & how we know this

AP Precalculus Course and Exam Description — College Board (2023)