Answer:
[tex]y=-8(x+5)^2-7[/tex]
Step-by-step explanation:
Given:
Focus: (-5,-9)
Directrix: y = -5
Find: equation of the parabola
The equation of the line perpendicular to the directrix and passing through the focus is x = -5. This is the line of parabola's symmetry.
The distance between the focus and the directrix is the parabola's parameter, so
[tex]p=|-9-(-5)|=|-9+5|=|-4|=4[/tex]
The vertex of the parabola lies on the line of symmetry and divides the distance between the focus and the directrix into two equal parts. So, its coordinates are (-5,-7).
Parabola goes in negative y-direction, thus, the equation of the parabola is
[tex]y-(-7)=-2\cdot 4(x-(-5))^2\\ \\y=-8(x+5)^2-7[/tex]
