Write the equation in vertex form for the parabola with focus $$ (0,10) $$ and directrix $$ y=6 $$. Simplify any fractions.

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To find the equation of the parabola in vertex form given the focus $$ (0,10) $$ and the directrix $$ y=6 $$, follow these steps:

1. **Determine the Vertex:**
   - The vertex lies exactly midway between the focus and the directrix.
   - The y-coordinate of the vertex is $$ \frac{10 + 6}{2} = 8 $$.
   - Thus, the vertex is at $$ (0, 8) $$.

2. **Find the Value of $$ p $$:**
   - $$ p $$ is the distance from the vertex to the focus.
   - $$ p = 10 - 8 = 2 $$.

3. **Determine the Orientation:**
   - Since the focus is above the directrix, the parabola opens **upward**.

4. **Use the Vertex Form Equation:**
   - The vertex form of a parabola that opens upward is:
     $$
     y = a(x - h)^2 + k
     $$
     where $$ (h, k) $$ is the vertex.
   - Here, $$ h = 0 $$ and $$ k = 8 $$, so:
     $$
     y = a x^2 + 8
     $$

5. **Calculate $$ a $$:**
   - The value of $$ a $$ is related to $$ p $$ by the formula $$ a = \frac{1}{4p} $$.
   - Substituting $$ p = 2 $$:
     $$
     a = \frac{1}{4 \times 2} = \frac{1}{8}
     $$

6. **Write the Final Equation:**
   - Substitute $$ a $$ back into the vertex form:
     $$
     y = \frac{1}{8}x^2 + 8
     $$

**Final Answer:**
$$
y = \frac{1}{8}\,x^{2} + 8
$$
The equation of the parabola in vertex form is $$ y = \frac{1}{8}x^{2} + 8 $$.

Write the equation in vertex form for the parabola with focus \( (0,10) \) and directrix \( y=6 \). Simplify any fractions.

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