To find the vertex form of the equation of a parabola given the directrix [tex]\( x = 6 \)[/tex] and the focus [tex]\( (3, -5) \)[/tex], we need to follow these detailed steps:
1. Identify the vertex: The vertex is the midpoint between the focus and the directrix.
- The x-coordinate of the vertex is the midpoint of [tex]\( 6 \)[/tex] (directrix x-coordinate) and [tex]\( 3 \)[/tex] (focus x-coordinate):
[tex]\[
\text{Vertex}_x = \frac{3 + 6}{2} = \frac{9}{2} = 4.5
\][/tex]
- The y-coordinate of the vertex remains the same as the y-coordinate of the focus, which is [tex]\( -5 \)[/tex].
[tex]\[
\text{Vertex coordinates} = (4.5, -5)
\][/tex]
2. Calculate the distance [tex]\( p \)[/tex] between the vertex and the focus (or directrix). This distance is crucial for determining the parameter in the equation of the parabola:
[tex]\[
p = |4.5 - 6| = 1.5
\][/tex]
3. Determine the constant [tex]\( a \)[/tex] in the vertex form of the parabola: The vertex form of the equation is given by [tex]\( x = a(y - k)^2 + h \)[/tex], where [tex]\( a = \frac{1}{4p} \)[/tex]:
[tex]\[
a = \frac{1}{4 \times 1.5} = \frac{1}{6} ≈ 0.16666666666666666
\][/tex]
So, the vertex form of the equation of the parabola is:
[tex]\[
x = \frac{1}{6} (y + 5)^2 + 4.5
\][/tex]
Therefore, filling in the boxes, the vertex form of the equation is:
[tex]\[
x = 0.16666666666666666(y + 5)^2 + 4.5
\][/tex]
So the correct answers to fill in the boxes are:
[tex]\[
x =\ \boxed{0.16666666666666666(y + } \ \boxed{5)}^2+ \ \boxed{4.5}
\][/tex]
