Find a parameterization for a circle of radius 7 with center $$ (2,5,-4) $$ in a plane parallel to the xy plane. Write your parameterization so the $$ x $$ component includes a positive cosine. $$ x(t)=\square $$ $$ y(t)=\square $$ $$ z(t)=\square $$

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To parameterize a circle of radius $$ r = 7 $$ centered at the point $$ (2, 5, -4) $$ in a plane parallel to the xy-plane, we can use the following steps:

1. **Identify the center and radius**: The center of the circle is $$ (h, k, l) = (2, 5, -4) $$ and the radius is $$ r = 7 $$.

2. **Parameterization of the circle**: The general parameterization of a circle in the xy-plane is given by:
   $$
   x(t) = h + r \cos(t)
   $$
   $$
   y(t) = k + r \sin(t)
   $$
   The z-coordinate remains constant since the circle lies in a plane parallel to the xy-plane:
   $$
   z(t) = l
   $$

3. **Substituting the values**:
   - For $$ x(t) $$:
     $$
     x(t) = 2 + 7 \cos(t)
     $$
   - For $$ y(t) $$:
     $$
     y(t) = 5 + 7 \sin(t)
     $$
   - For $$ z(t) $$:
     $$
     z(t) = -4
     $$

Thus, the parameterization of the circle is:

$$
x(t) = 2 + 7 \cos(t)
$$
$$
y(t) = 5 + 7 \sin(t)
$$
$$
z(t) = -4
$$

So, the final answer is:
- $$ x(t) = 2 + 7 \cos(t) $$
- $$ y(t) = 5 + 7 \sin(t) $$
- $$ z(t) = -4 $$
$$ x(t) = 2 + 7 \cos(t) $$ $$ y(t) = 5 + 7 \sin(t) $$ $$ z(t) = -4 $$

Find a parameterization for a circle of radius 7 with center \( (2,5,-4) \) in a plane parallel to the xy plane. Write your parameterization so the \( x \) component includes a positive cosine. \( x(t)=\square \) \( y(t)=\square \) \( z(t)=\square \)

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