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: 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 2: 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 3: 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 4: 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 5: Convert $x = 1 + t, \, y = 2t$ to its Cartesian form.

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

Flashcard 6: What is the parametric form of a vertical line $x = c$?

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

Flashcard 7: 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 8: 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 9: 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 10: 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 11: Find $x$ when $t = 0$ for $x = 4 + 2t, \, y = 3 - t$.

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

Flashcard 12: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: 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 19: Convert $x = t, \, y = 2t + 1$ to its Cartesian form.

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

Flashcard 20: 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 21: Find $x$ when $t = 2$ for $x = 1 + 3t, \, y = 2t$.

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

Flashcard 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: 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 31: Find $y$ when $t = 1$ for $x = 3 + 2t, \, y = 4 - t$.

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

Flashcard 32: 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 33: 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 34: 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 35: 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 36: 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 37: 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.