Given:
The focus of the parabola is at (-4,8).
The directrix is at x=-6.
To find:
The equation of the parabola.
Solution:
The directrix is at x=-6, which is a vertical line. So, the parabola is horizontal.
The equation of a horizontal parabola is
[tex](y-k)^2=4p(x-h)[/tex]
Where, (h,k) is vertex, (h+p,k) is focus and x=h-p.
The focus of the parabola is at (-4,8).
[tex](h+p,k)=(-4,8)[/tex]
[tex]h+p=-4[/tex] ...(i)
[tex]k=8[/tex]
The directrix is at x=-6.
[tex]h-p=-6[/tex] ...(ii)
Adding (i) and (ii), we get
[tex]2h=-10[/tex]
[tex]h=-5[/tex]
Putting h=-5 in (i), we get
[tex]-5+p=-4[/tex]
[tex]p=-4+5[/tex]
[tex]p=1[/tex]
Putting h=-5, k=8 and p=1 in the standard form of the parabola.
[tex](y-8)^2=4(1)(x-(-5))[/tex]
[tex](y-8)^2=4(x+5)[/tex]
Therefore, the required equation of the parabola is [tex](y-8)^2=4(x+5)[/tex].
