Parametrically Defined Circles and Lines - AP Precalculus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

22. What is the parametric form of a circle centered at the origin with radius $r$?

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

23. What is the parametric form of a circle with center $(0, 0)$ and radius $r$?

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

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

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

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

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

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

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

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

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

28. What is the parametric form of a line with slope $m$ passing through $(0, 0)$?

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

29. What are the parametric equations for a line through $(x_1, y_1)$ with slope $m$?

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

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

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

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

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

32. What is the parametric form of a horizontal line $y = c$?

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

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

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

34. What are the parametric equations for a line through $(0, 0)$ with slope $m$?

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

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

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

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

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

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

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

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