Parametrically Defined Circles and Lines - AP Precalculus
Card 1 of 30
Identify the $t$ value for the point $(r, 0)$ on $x = r , \cos(t), , y = r , \sin(t)$.
Identify the $t$ value for the point $(r, 0)$ on $x = r , \cos(t), , y = r , \sin(t)$.
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$t = 0$. At $t = 0$, $\cos(0) = 1$ and $\sin(0) = 0$.
$t = 0$. At $t = 0$, $\cos(0) = 1$ and $\sin(0) = 0$.
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What is the parametric form of a vertical line $x = c$?
What is the parametric form of a vertical line $x = c$?
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$x = c, , y = t$. Parameter $t$ varies vertically, $x$ remains constant.
$x = c, , y = t$. Parameter $t$ varies vertically, $x$ remains constant.
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What is the parametric form of a horizontal line $y = c$?
What is the parametric form of a horizontal line $y = c$?
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$x = t, , y = c$. Parameter $t$ varies along the line, $y$ remains constant.
$x = t, , y = c$. Parameter $t$ varies along the line, $y$ remains constant.
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Convert $x = 2 + 3t, , y = 4 + 5t$ to its Cartesian form.
Convert $x = 2 + 3t, , y = 4 + 5t$ to its Cartesian form.
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$y = 4 + \frac{5}{3}(x - 2)$. Eliminate parameter: $t = \frac{x-2}{3}$, substitute into $y$.
$y = 4 + \frac{5}{3}(x - 2)$. Eliminate parameter: $t = \frac{x-2}{3}$, substitute into $y$.
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What is the parametric form of a circle with center $(h, k)$ and radius $r$?
What is the parametric form of a circle with center $(h, k)$ and radius $r$?
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$x = h + r , \cos(t), , y = k + r , \sin(t)$. Translation of circle center from origin to $(h, k)$.
$x = h + r , \cos(t), , y = k + r , \sin(t)$. Translation of circle center from origin to $(h, k)$.
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Find the $x$-coordinate when $t = \pi$ for $x = 2 + 3t, , y = 4 - t$.
Find the $x$-coordinate when $t = \pi$ for $x = 2 + 3t, , y = 4 - t$.
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$x = 2 + 3\pi$. Substitute $t = \pi$ into the $x$ equation.
$x = 2 + 3\pi$. Substitute $t = \pi$ into the $x$ equation.
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Identify the parameter $t$ value at the topmost point of $x = r , \cos(t), y = r , \sin(t)$.
Identify the parameter $t$ value at the topmost point of $x = r , \cos(t), y = r , \sin(t)$.
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$t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.
$t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.
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Find the $x$-coordinate when $t = \frac{\theta}{2}$ for $x = 5 , \cos(t), , y = 5 , \sin(t)$.
Find the $x$-coordinate when $t = \frac{\theta}{2}$ for $x = 5 , \cos(t), , y = 5 , \sin(t)$.
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$x = 5 , \cos\left(\frac{\theta}{2}\right)$. Substitute $t = \frac{\theta}{2}$ into the $x$ equation.
$x = 5 , \cos\left(\frac{\theta}{2}\right)$. Substitute $t = \frac{\theta}{2}$ into the $x$ equation.
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What are the parametric equations for a line through $(x_1, y_1)$ with slope $m$?
What are the parametric equations for a line through $(x_1, y_1)$ with slope $m$?
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$x = x_1 + t, , y = y_1 + mt$. Direction vector $(1, m)$ from point-slope form.
$x = x_1 + t, , y = y_1 + mt$. Direction vector $(1, m)$ from point-slope form.
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What is the parametric form of a circle centered at the origin with radius $r$?
What is the parametric form of a circle centered at the origin with radius $r$?
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$x = r , \cos(t), , y = r , \sin(t)$. Standard circular parametrization using trigonometric functions.
$x = r , \cos(t), , y = r , \sin(t)$. Standard circular parametrization using trigonometric functions.
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What is the parametric form for a line parallel to $y = 3x + 2$ through $(4, 1)$?
What is the parametric form for a line parallel to $y = 3x + 2$ through $(4, 1)$?
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$x = 4 + t, , y = 1 + 3t$. Parallel lines have same slope; direction vector $(1, 3)$.
$x = 4 + t, , y = 1 + 3t$. Parallel lines have same slope; direction vector $(1, 3)$.
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Determine the $t$ value for $(0, r)$ on $x = r , \cos(t), , y = r , \sin(t)$.
Determine the $t$ value for $(0, r)$ on $x = r , \cos(t), , y = r , \sin(t)$.
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$t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.
$t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.
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Identify the $t$ value for the point $(-r, 0)$ on $x = r , \cos(t), , y = r , \sin(t)$.
Identify the $t$ value for the point $(-r, 0)$ on $x = r , \cos(t), , y = r , \sin(t)$.
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$t = \pi$. At $t = \pi$, $\cos(\pi) = -1$ and $\sin(\pi) = 0$.
$t = \pi$. At $t = \pi$, $\cos(\pi) = -1$ and $\sin(\pi) = 0$.
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Convert $x = 2t, , y = 3 + t$ to its Cartesian form.
Convert $x = 2t, , y = 3 + t$ to its Cartesian form.
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$y = 3 + \frac{x}{2}$. Eliminate $t$: $t = \frac{x}{2}$, substitute into $y$.
$y = 3 + \frac{x}{2}$. Eliminate $t$: $t = \frac{x}{2}$, substitute into $y$.
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Convert $x = 2 + 3t, , y = 4 + 5t$ to its Cartesian form.
Convert $x = 2 + 3t, , y = 4 + 5t$ to its Cartesian form.
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$y = 4 + \frac{5}{3}(x - 2)$. Eliminate parameter: $t = \frac{x-2}{3}$, substitute into $y$.
$y = 4 + \frac{5}{3}(x - 2)$. Eliminate parameter: $t = \frac{x-2}{3}$, substitute into $y$.
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Convert $x = t, , y = 2t + 1$ to its Cartesian form.
Convert $x = t, , y = 2t + 1$ to its Cartesian form.
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$y = 2x + 1$. Direct substitution since $x = t$.
$y = 2x + 1$. Direct substitution since $x = t$.
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Identify the $t$ value for the point $(0, r)$ on $x = r , \cos(t), , y = r , \sin(t)$.
Identify the $t$ value for the point $(0, r)$ on $x = r , \cos(t), , y = r , \sin(t)$.
Tap to reveal answer
$t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.
$t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.
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What is the parametric form of a line with slope $m$ passing through $(0, 0)$?
What is the parametric form of a line with slope $m$ passing through $(0, 0)$?
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$x = t, , y = mt$. Line through origin with direction vector $(1, m)$.
$x = t, , y = mt$. Line through origin with direction vector $(1, m)$.
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Convert $x = 1 + t, , y = 2t$ to its Cartesian form.
Convert $x = 1 + t, , y = 2t$ to its Cartesian form.
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$y = 2(x - 1)$. Eliminate $t$: $t = x - 1$, substitute into $y$.
$y = 2(x - 1)$. Eliminate $t$: $t = x - 1$, substitute into $y$.
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Determine the $y$-coordinate at $t = 0$ for $x = 3 , \cos(t), , y = 3 , \sin(t)$.
Determine the $y$-coordinate at $t = 0$ for $x = 3 , \cos(t), , y = 3 , \sin(t)$.
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$y = 0$. At $t = 0$, $\sin(0) = 0$ and $\cos(0) = 1$.
$y = 0$. At $t = 0$, $\sin(0) = 0$ and $\cos(0) = 1$.
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Determine the $t$ value for $(0, r)$ on $x = r , \cos(t), , y = r , \sin(t)$.
Determine the $t$ value for $(0, r)$ on $x = r , \cos(t), , y = r , \sin(t)$.
Tap to reveal answer
$t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.
$t = \frac{\pi}{2}$. At $t = \frac{\pi}{2}$, $\cos(t) = 0$ and $\sin(t) = 1$.
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What is the parametric form of a circle with center $(h, k)$ and radius $r$?
What is the parametric form of a circle with center $(h, k)$ and radius $r$?
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$x = h + r , \cos(t), , y = k + r , \sin(t)$. Translation of circle center from origin to $(h, k)$.
$x = h + r , \cos(t), , y = k + r , \sin(t)$. Translation of circle center from origin to $(h, k)$.
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Convert $x = 3t, , y = 4 - t$ to its Cartesian form.
Convert $x = 3t, , y = 4 - t$ to its Cartesian form.
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$y = 4 - \frac{1}{3}x$. Eliminate $t$: $t = \frac{x}{3}$, substitute into $y$.
$y = 4 - \frac{1}{3}x$. Eliminate $t$: $t = \frac{x}{3}$, substitute into $y$.
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What are the parametric equations for a line through $(x_0, y_0)$ with direction vector $(a, b)$?
What are the parametric equations for a line through $(x_0, y_0)$ with direction vector $(a, b)$?
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$x = x_0 + at, , y = y_0 + bt$. Direction vector $(a, b)$ from point $(x_0, y_0)$.
$x = x_0 + at, , y = y_0 + bt$. Direction vector $(a, b)$ from point $(x_0, y_0)$.
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Determine the $t$ value for the point $(-r, 0)$ on $x = r , \cos(t), , y = r , \sin(t)$.
Determine the $t$ value for the point $(-r, 0)$ on $x = r , \cos(t), , y = r , \sin(t)$.
Tap to reveal answer
$t = \pi$. At $t = \pi$, $\cos(\pi) = -1$ and $\sin(\pi) = 0$.
$t = \pi$. At $t = \pi$, $\cos(\pi) = -1$ and $\sin(\pi) = 0$.
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Convert $x = 3t, , y = 4 - t$ to its Cartesian form.
Convert $x = 3t, , y = 4 - t$ to its Cartesian form.
Tap to reveal answer
$y = 4 - \frac{1}{3}x$. Eliminate $t$: $t = \frac{x}{3}$, substitute into $y$.
$y = 4 - \frac{1}{3}x$. Eliminate $t$: $t = \frac{x}{3}$, substitute into $y$.
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Find $x$ when $t = 0$ for $x = 4 + 2t, , y = 3 - t$.
Find $x$ when $t = 0$ for $x = 4 + 2t, , y = 3 - t$.
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$x = 4$. Substitute $t = 0$ into the $x$ equation.
$x = 4$. Substitute $t = 0$ into the $x$ equation.
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What is the parametric form for a line parallel to $y = 3x + 2$ through $(4, 1)$?
What is the parametric form for a line parallel to $y = 3x + 2$ through $(4, 1)$?
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$x = 4 + t, , y = 1 + 3t$. Parallel lines have same slope; direction vector $(1, 3)$.
$x = 4 + t, , y = 1 + 3t$. Parallel lines have same slope; direction vector $(1, 3)$.
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Find $x$ when $t = 2$ for $x = 1 + 3t, , y = 2t$.
Find $x$ when $t = 2$ for $x = 1 + 3t, , y = 2t$.
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$x = 7$. Substitute $t = 2$ into the $x$ equation.
$x = 7$. Substitute $t = 2$ into the $x$ equation.
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Find the Cartesian equation of $x = 5 , \cos(t), , y = 5 , \sin(t)$.
Find the Cartesian equation of $x = 5 , \cos(t), , y = 5 , \sin(t)$.
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$x^2 + y^2 = 25$. Use identity $\cos^2(t) + \sin^2(t) = 1$ with radius 5.
$x^2 + y^2 = 25$. Use identity $\cos^2(t) + \sin^2(t) = 1$ with radius 5.
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