We are asked to write the equation of parabola with focus [tex]F(-1,-13)[/tex] and directrix [tex]y=-9[/tex].
Let us say that point (x,y) is on parabola.
We know that any point on parabola is equidistant from focus and directrix. Using distance formula we will get:
[tex]\sqrt{(y-(-9))^2}=\sqrt{(x-(-1))^2+(y-(-13))^2}[/tex]
[tex]\sqrt{(y+9)^2}=\sqrt{(x+1)^2+(y+13)^2}[/tex]
Square both sides:
[tex](y+9)^2=(x+1)^2+(y+13)^2[/tex]
[tex]y^2+18y+81=(x+1)^2+y^2+26y+169[/tex]
[tex]18y+81-81=(x+1)^2+26y+169-81[/tex]
[tex]18y=(x+1)^2+26y+88[/tex]
[tex]18y-26y=(x+1)^2+26y-26y+88[/tex]
[tex]-8y=(x+1)^2+88[/tex]
[tex]\frac{-8y}{-8}=\frac{(x+1)^2+88}{-8}[/tex]
[tex]y=-\frac{(x+1)^2}{8}-11[/tex]
Therefore, our required equation would be [tex]y=-\frac{(x+1)^2}{8}-11[/tex].
