Answer:
[tex](x+8)^2=12(y-3)[/tex]
Step-by-step explanation:
As the directrix is horizontal (y = 0), the parabola is vertical.
The standard formula of a vertical parabola is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Standard form of a vertical parabola}}\\\\(x-h)^2=4p(y-k)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{Vertex:\;$(h,k)$}\\\phantom{ww}\bullet\;\textsf{Focus:\;$(h,k + p)$}\\\phantom{ww}\bullet\;\textsf{Directrix:\;$y=(k-p)$}\\\phantom{ww}\bullet\;\textsf{Axis of symmetry:\;$x=h$}\\\phantom{ww}\bullet\;\textsf{$p =$ distance between vertex and focus.}\end{array}}[/tex]
The vertex of a vertical parabola shares the same x-coordinate as the focus and lies equidistant between the focus and the directrix.
Given that the focus is at (-8, 6) and the directrix is y = 0, the coordinates of the vertex are:
[tex](h,k)=(-8, 3)[/tex]
The value of p is the distance between the vertex and the focus.
Given that the focus is (-8, 6) and the vertex is (-8, 3), then p is the difference between the y-coordinates:
[tex]p = 6 - 3 = 3[/tex]
To write the equation of the parabola, substitute h = -8, k = 3 and p = 3 into the formula and simplify:
[tex](x - (-8))^2=4\cdot 3(y-3)\\\\\\(x+8)^2=12(y-3)[/tex]
Therefore, the equation of the parabola is:
[tex]\Large\boxed{\boxed{(x+8)^2=12(y-3)}}[/tex]
