To determine the vertex form of the equation of the parabola given the directrix [tex]\( x = 6 \)[/tex] and the focus [tex]\( (3, -5) \)[/tex], follow these steps:
1. Calculate the Vertex:
The vertex is the midpoint between the focus and the directrix. The x-coordinate of the vertex (which lies on the parabola's axis of symmetry) can be calculated as:
[tex]\[
\text{vertex}_x = \frac{\text{focus}_x + \text{directrix}_x}{2} = \frac{3 + 6}{2} = 4.5
\][/tex]
Since the vertex lies on the axis of symmetry of the parabola, its y-coordinate will be the same as the focus's y-coordinate:
[tex]\[
\text{vertex}_y = -5
\][/tex]
2. Determine the value of [tex]\( p \)[/tex]:
The value of [tex]\( p \)[/tex] is the distance from the vertex to the directrix:
[tex]\[
p = \text{directrix}_x - \text{vertex}_x = 6 - 4.5 = 1.5
\][/tex]
3. Write the vertex form of the equation:
The general vertex form of a parabola that opens horizontally is:
[tex]\[
x = h + \frac{1}{4p} (y - k)^2
\][/tex]
Here, [tex]\( (h, k) \)[/tex] is the vertex. Substituting the values we have:
[tex]\[
x = 4.5 + \frac{1}{4 \times 1.5} (y + 5)^2
\][/tex]
4. Simplify the equation:
Simplify the term involving [tex]\( 4p \)[/tex]:
[tex]\[
4p = 4 \times 1.5 = 6
\][/tex]
So, the equation becomes:
[tex]\[
x = 4.5 + \frac{1}{6} (y + 5)^2
\][/tex]
Therefore, the vertex form of the equation of the parabola is:
[tex]\[
x = 4.5 - \frac{1}{6} (y + 5)^2
\][/tex]
Finally, fill in the blanks in the provided equation:
[tex]\[
x = \boxed{4.5} - \frac{1}{6} (y + \boxed{5})^2
\][/tex]
