Answer:
[tex]-\frac{1}{2}y^2-4y+1=x[/tex]
Step-by-step explanation:
A parabola has a focus of F(8.5,−4) and a directrix of x=9.5.
General form of horizontal parabola is
[tex](y-k)^2=4p(x-h)[/tex]
the distance between directrix and focus is the value of p
so p = 8.5 - 9.5 = -0.5
Focus is (h+p , k), given focus is (8.5, -4)
So k = -4 and h+p = 8.5
we know p = -0.5
h +p = 8.5
h - 0.5 = 8.5
so h= 9 and k = -4
vertex is (h,k) that is (9, -4)
Now plug in the value in the general equation
[tex](y-k)^2=4p(x-h)[/tex], k= -4, h= 9 , p = -0.5
[tex](y+4)^2=4(-0.5)(x-9)[/tex]
[tex](y+4)^2=-2(x-9)[/tex]
[tex]y^2+8y+16=-2x+18[/tex]
subtract 18 on both sides
[tex]y^2+8y-2=-2x[/tex]
Divide whole equation by -2
[tex]-\frac{1}{2}y^2-4y+1=x[/tex]
