To find the vertex form of the equation of the parabola given the focus [tex]\((3, -5)\)[/tex] and the directrix [tex]\(x = 6\)[/tex], follow these steps:
1. Identify the coordinates of the vertex [tex]\((h, k)\)[/tex]. Since the focus is [tex]\((3, -5)\)[/tex] and the directrix is [tex]\(x = 6\)[/tex], the vertex [tex]\((h, k)\)[/tex] lies midway between the focus and the directrix along the x-axis. Given these specific values, the vertex is [tex]\((3, -5)\)[/tex].
2. Determine the value of [tex]\(a\)[/tex] (the coefficient in the vertex form of the parabola equation). We use the formula [tex]\(a = \frac{1}{2(h - \text{directrix})}\)[/tex].
- [tex]\(h = 3\)[/tex]
- The directrix is [tex]\(6\)[/tex]
3. Compute the value of [tex]\(a\)[/tex]:
[tex]\[
a = \frac{1}{2(3 - 6)} = \frac{1}{2(-3)} = -\frac{1}{6}
\][/tex]
4. Using the vertex form of the parabola equation with horizontal orientation:
[tex]\[
x = a(y - k)^2 + h
\][/tex]
Substitute in the calculated values:
[tex]\[
x = -\frac{1}{6}(y - (-5))^2 + 3
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
x = -\frac{1}{6}(y + 5)^2 + 3
\][/tex]
Thus, the vertex form of the equation of the parabola is:
[tex]\[
x = -\frac{1}{6}(y + 5)^2 + 3
\][/tex]
Therefore, the correct answers to fill in the boxes are:
[tex]\[
x = \boxed{-\frac{1}{6}}(y + \boxed{5})^2 + \boxed{3}
\][/tex]
