AP Precalculus Flashcards: Parametrically Defined Circles And Lines

What this deck covers

This deck focuses on Parametrically Defined Circles And Lines, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Determine the $y$-coordinate when $t = \pi$ for $x = 2 + 3t, \, y = 4 - t$.

Answer: $y = 4 - \pi$. Substitute $t = \pi$ into the $y$ equation.

Flashcard 2: Identify the $t$ value for the point $(-r, 0)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = \pi$. At $t = \pi$, $\cos(\pi) = -1$ and $\sin(\pi) = 0$.

Flashcard 3: What is the parametric form of a circle with center $(h, k)$ and radius $r$?

Answer: $x = h + r \, \cos(t), \, y = k + r \, \sin(t)$. Translation of circle center from origin to $(h, k)$.

Flashcard 4: Convert $x = 1 + 2t, \, y = 3t$ to its Cartesian form.

Answer: $y = \frac{3}{2}(x - 1)$. Eliminate $t$: $t = \frac{x-1}{2}$, substitute into $y$.

Flashcard 5: Find the $y$-coordinate when $t = \frac{\pi}{4}$ for $x = 4 \, \cos(t), \, y = 4 \, \sin(t)$.

Answer: $y = 4 \, \sin\left(\frac{\pi}{4}\right)$. Substitute $t = \frac{\pi}{4}$ into the $y$ equation.

Flashcard 6: Convert $x = 2t, \, y = 3 + t$ to its Cartesian form.

Answer: $y = 3 + \frac{x}{2}$. Eliminate $t$: $t = \frac{x}{2}$, substitute into $y$.

Flashcard 7: Find the Cartesian equation of $x = 5 \, \cos(t), \, y = 5 \, \sin(t)$.

Answer: $x^2 + y^2 = 25$. Use identity $\cos^2(t) + \sin^2(t) = 1$ with radius 5.

Flashcard 8: Determine the $t$ value for the point $(-r, 0)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = \pi$. At $t = \pi$, $\cos(\pi) = -1$ and $\sin(\pi) = 0$.

Flashcard 9: What is the parametric form for a line parallel to $y = 3x + 2$ through $(4, 1)$?

Answer: $x = 4 + t, \, y = 1 + 3t$. Parallel lines have same slope; direction vector $(1, 3)$.

Flashcard 10: Convert $x = 1 + t, \, y = 2t$ to its Cartesian form.

Answer: $y = 2(x - 1)$. Eliminate $t$: $t = x - 1$, substitute into $y$.

Flashcard 11: Find $y$ when $t = 1$ for $x = 3 + 2t, \, y = 4 - t$.

Answer: $y = 3$. Substitute $t = 1$ into the $y$ equation.

Flashcard 12: Identify the $t$ value for the point $(0, -r)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = \frac{3\pi}{2}$. At $t = \frac{3\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = -1$.

Flashcard 13: What is the parametric form of a vertical line $x = c$?

Answer: $x = c, \, y = t$. Parameter $t$ varies vertically, $x$ remains constant.

Flashcard 14: Determine the $y$-coordinate at $t = 0$ for $x = 3 \, \cos(t), \, y = 3 \, \sin(t)$.

Answer: $y = 0$. At $t = 0$, $\sin(0) = 0$ and $\cos(0) = 1$.

Flashcard 15: Convert $x = t, \, y = 2t + 1$ to its Cartesian form.

Answer: $y = 2x + 1$. Direct substitution since $x = t$.

Flashcard 16: Find the $x$-coordinate when $t = \frac{\theta}{2}$ for $x = 5 \, \cos(t), \, y = 5 \, \sin(t)$.

Answer: $x = 5 \, \cos\left(\frac{\theta}{2}\right)$. Substitute $t = \frac{\theta}{2}$ into the $x$ equation.

Flashcard 17: Find $x$ when $t = 0$ for $x = 4 + 2t, \, y = 3 - t$.

Answer: $x = 4$. Substitute $t = 0$ into the $x$ equation.

Flashcard 18: Find the $x$-coordinate when $t = \pi$ for $x = 2 + 3t, \, y = 4 - t$.

Answer: $x = 2 + 3\pi$. Substitute $t = \pi$ into the $x$ equation.

Flashcard 19: What is the parametric form of a circle with center $(0, 0)$ and radius $r$?

Answer: $x = r \, \cos(t), \, y = r \, \sin(t)$. Standard circular parametrization centered at origin.

Flashcard 20: Convert $x = 3t, \, y = 4 - t$ to its Cartesian form.

Answer: $y = 4 - \frac{1}{3}x$. Eliminate $t$: $t = \frac{x}{3}$, substitute into $y$.

Flashcard 21: Identify the $t$ value for the point $(0, r)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.

Flashcard 22: Identify the $t$ value for the point $(0, r)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.

Flashcard 23: Find $x$ when $t = 0$ for $x = 4 + 2t, \, y = 3 - t$.

Answer: $x = 4$. Substitute $t = 0$ into the $x$ equation.

Flashcard 24: Determine the $t$ value for the point $(-r, 0)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = \pi$. At $t = \pi$, $\cos(\pi) = -1$ and $\sin(\pi) = 0$.

Flashcard 25: Find the $x$-coordinate when $t = \pi$ for $x = 2 + 3t, \, y = 4 - t$.

Answer: $x = 2 + 3\pi$. Substitute $t = \pi$ into the $x$ equation.

Flashcard 26: What is the parametric form of a line through $(a, b)$ parallel to $y = mx + c$?

Answer: $x = a + t, \, y = b + mt$. Direction vector from slope $m$ of parallel line.

Flashcard 27: Identify the $t$ value for the point $(0, -r)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = \frac{3\pi}{2}$. At $t = \frac{3\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = -1$.

Flashcard 28: Determine the $t$ value for $(0, r)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.

Flashcard 29: What is the parametric form of a line through $(a, b)$ parallel to $y = mx + c$?

Answer: $x = a + t, \, y = b + mt$. Direction vector from slope $m$ of parallel line.

Flashcard 30: Find the Cartesian equation of $x = 1 + 2t, \, y = 3 - t$.

Answer: $y = 3 - \frac{1}{2}(x - 1)$. Eliminate $t$: $t = \frac{x-1}{2}$, substitute into $y$.

Flashcard 31: Convert $x = 2 + 3t, \, y = 4 + 5t$ to its Cartesian form.

Answer: $y = 4 + \frac{5}{3}(x - 2)$. Eliminate parameter: $t = \frac{x-2}{3}$, substitute into $y$.

Flashcard 32: Convert $x = t, \, y = 2t + 1$ to its Cartesian form.

Answer: $y = 2x + 1$. Direct substitution since $x = t$.

Flashcard 33: Find the $y$-coordinate when $t = \frac{\pi}{4}$ for $x = 4 \, \cos(t), \, y = 4 \, \sin(t)$.

Answer: $y = 4 \, \sin\left(\frac{\pi}{4}\right)$. Substitute $t = \frac{\pi}{4}$ into the $y$ equation.

Flashcard 34: What is the parametric form of a vertical line $x = c$?

Answer: $x = c, \, y = t$. Parameter $t$ varies vertically, $x$ remains constant.

Flashcard 35: Find $x$ when $t = 2$ for $x = 1 + 3t, \, y = 2t$.

Answer: $x = 7$. Substitute $t = 2$ into the $x$ equation.

Flashcard 36: What is the parametric form of a circle centered at the origin with radius $r$?

Answer: $x = r \, \cos(t), \, y = r \, \sin(t)$. Standard circular parametrization using trigonometric functions.

Flashcard 37: Convert $x = 4 \, \cos(t), \, y = 4 \, \sin(t)$ to its Cartesian form.

Answer: $x^2 + y^2 = 16$. Use identity $\cos^2(t) + \sin^2(t) = 1$ with radius 4.

Flashcard 38: What is the parametric form of a circle centered at the origin with radius $r$?

Answer: $x = r \, \cos(t), \, y = r \, \sin(t)$. Standard circular parametrization using trigonometric functions.

Flashcard 39: Find the Cartesian equation of $x = 5 \, \cos(t), \, y = 5 \, \sin(t)$.

Answer: $x^2 + y^2 = 25$. Use identity $\cos^2(t) + \sin^2(t) = 1$ with radius 5.

Flashcard 40: What is the parametric form of a circle with center $(0, 0)$ and radius $r$?

Answer: $x = r \, \cos(t), \, y = r \, \sin(t)$. Standard circular parametrization centered at origin.

Flashcard 41: What are the parametric equations for a line through $(0, 0)$ with slope $m$?

Answer: $x = t, \, y = mt$. Line through origin with direction vector $(1, m)$.

Flashcard 42: Convert $x = 2t, \, y = 3 + t$ to its Cartesian form.

Answer: $y = 3 + \frac{x}{2}$. Eliminate $t$: $t = \frac{x}{2}$, substitute into $y$.

Flashcard 43: Convert $x = 2 + 4t, \, y = 3 + 2t$ to its Cartesian form.

Answer: $y = 3 + \frac{1}{2}(x - 2)$. Eliminate $t$: $t = \frac{x-2}{4}$, substitute into $y$.

Flashcard 44: Identify the parameter $t$ value at the topmost point of $x = r \, \cos(t), y = r \, \sin(t)$.

Answer: $t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.

Flashcard 45: What is the parametric form of a circle with center $(h, k)$ and radius $r$?

Answer: $x = h + r \, \cos(t), \, y = k + r \, \sin(t)$. Translation of circle center from origin to $(h, k)$.

Flashcard 46: Convert $x = 3t, \, y = 4 - t$ to its Cartesian form.

Answer: $y = 4 - \frac{1}{3}x$. Eliminate $t$: $t = \frac{x}{3}$, substitute into $y$.

Flashcard 47: Convert $x = 1 + 2t, \, y = 3t$ to its Cartesian form.

Answer: $y = \frac{3}{2}(x - 1)$. Eliminate $t$: $t = \frac{x-1}{2}$, substitute into $y$.

Flashcard 48: Identify the $t$ value for the point $(0, -r)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = \frac{3\pi}{2}$. At $t = \frac{3\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = -1$.

Flashcard 49: What is the parametric form of a horizontal line $y = c$?

Answer: $x = t, \, y = c$. Parameter $t$ varies along the line, $y$ remains constant.

Flashcard 50: Convert $x = 1 + t, \, y = 2t$ to its Cartesian form.

Answer: $y = 2(x - 1)$. Eliminate $t$: $t = x - 1$, substitute into $y$.

Flashcard 51: What is the parametric form of a line with slope $m$ passing through $(0, 0)$?

Answer: $x = t, y = mt$. Line through origin with direction vector $(1, m)$.

Flashcard 52: What is the parametric form of a line with slope $m$ passing through $(0, 0)$?

Answer: $x = t, \, y = mt$. Line through origin with direction vector $(1, m)$.

Flashcard 53: Identify the $t$ value for the point $(-r, 0)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = \pi$. At $t = \pi$, $\cos(\pi) = -1$ and $\sin(\pi) = 0$.

Flashcard 54: Find $x$ when $t = 2$ for $x = 1 + 3t, \, y = 2t$.

Answer: $x = 7$. Substitute $t = 2$ into the $x$ equation.

Flashcard 55: Determine the $t$ value for $(0, r)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.

Flashcard 56: What $t$ value corresponds to the point $(h + r, k)$ on $x = h + r \, \cos(t), y = k + r \, \sin(t)$?

Answer: $t = 0$. At $t = 0$, $\cos(0) = 1$ gives rightmost point.

Flashcard 57: What is the parametric form for a line parallel to $y = 3x + 2$ through $(4, 1)$?

Answer: $x = 4 + t, \, y = 1 + 3t$. Parallel lines have same slope; direction vector $(1, 3)$.

Flashcard 58: What are the parametric equations for a line through $(x_1, y_1)$ with slope $m$?

Answer: $x = x_1 + t, \, y = y_1 + mt$. Direction vector $(1, m)$ from point-slope form.

Flashcard 59: Identify the $t$ value for the point $(r, 0)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = 0$. At $t = 0$, $\cos(0) = 1$ and $\sin(0) = 0$.

Flashcard 60: Find $y$ when $t = 1$ for $x = 3 + 2t, \, y = 4 - t$.

Answer: $y = 3$. Substitute $t = 1$ into the $y$ equation.

Flashcard 61: What is the parametric form of a horizontal line $y = c$?

Answer: $x = t, \, y = c$. Parameter $t$ varies along the line, $y$ remains constant.

Flashcard 62: Convert $x = 2 + 4t, \, y = 3 + 2t$ to its Cartesian form.

Answer: $y = 3 + \frac{1}{2}(x - 2)$. Eliminate $t$: $t = \frac{x-2}{4}$, substitute into $y$.

Flashcard 63: Identify the $t$ value for the point $(r, 0)$ on $x = r \, \cos(t), \, y = r \, \sin(t)$.

Answer: $t = 0$. At $t = 0$, $\cos(0) = 1$ and $\sin(0) = 0$.

Flashcard 64: What $t$ value corresponds to the point $(h + r, k)$ on $x = h + r \, \cos(t), y = k + r \, \sin(t)$?

Answer: $t = 0$. At $t = 0$, $\cos(0) = 1$ gives rightmost point.

Flashcard 65: Find the Cartesian equation of $x = 1 + 2t, \, y = 3 - t$.

Answer: $y = 3 - \frac{1}{2}(x - 1)$. Eliminate $t$: $t = \frac{x-1}{2}$, substitute into $y$.

Flashcard 66: What are the parametric equations for a line through $(0, 0)$ with slope $m$?

Answer: $x = t, \, y = mt$. Line through origin with direction vector $(1, m)$.

Flashcard 67: What are the parametric equations for a line through $(x_0, y_0)$ with direction vector $(a, b)$?

Answer: $x = x_0 + at, \, y = y_0 + bt$. Direction vector $(a, b)$ from point $(x_0, y_0)$.

Flashcard 68: Convert $x = 2 + 3t, \, y = 4 + 5t$ to its Cartesian form.

Answer: $y = 4 + \frac{5}{3}(x - 2)$. Eliminate parameter: $t = \frac{x-2}{3}$, substitute into $y$.

Flashcard 69: Determine the $y$-coordinate at $t = 0$ for $x = 3 \, \cos(t), \, y = 3 \, \sin(t)$.

Answer: $y = 0$. At $t = 0$, $\sin(0) = 0$ and $\cos(0) = 1$.

Flashcard 70: Find the $x$-coordinate when $t = \frac{\theta}{2}$ for $x = 5 \, \cos(t), \, y = 5 \, \sin(t)$.

Answer: $x = 5 \, \cos\left(\frac{\theta}{2}\right)$. Substitute $t = \frac{\theta}{2}$ into the $x$ equation.

Flashcard 71: What are the parametric equations for a line through $(x_0, y_0)$ with direction vector $(a, b)$?

Answer: $x = x_0 + at, \, y = y_0 + bt$. Direction vector $(a, b)$ from point $(x_0, y_0)$.

Flashcard 72: What are the parametric equations for a line through $(x_1, y_1)$ with slope $m$?

Answer: $x = x_1 + t, \, y = y_1 + mt$. Direction vector $(1, m)$ from point-slope form.

Flashcard 73: Convert $x = 4 \, \cos(t), \, y = 4 \, \sin(t)$ to its Cartesian form.

Answer: $x^2 + y^2 = 16$. Use identity $\cos^2(t) + \sin^2(t) = 1$ with radius 4.

Flashcard 74: Determine the $y$-coordinate when $t = \pi$ for $x = 2 + 3t, \, y = 4 - t$.

Answer: $y = 4 - \pi$. Substitute $t = \pi$ into the $y$ equation.