Find the focus of the parabola with the equation –y^{2}-3x-20y-92=0

To find the focus, convert this equation to standard form by completing the square:

–y^{2}-3x-20y-92=0

-3x-92=y^{2}+20y

-3x-92+100=y^{2}+20y+100

-3x+8=(y+10)^{2}

-3(x-8/3)=(y+10)^{2}

Comparing -3(x-8/3)=(y+10)^{2} to

(y-k)^{2}=4p(x-h)

k=-10

p=-3/4

h=8/3

The focus of the parabola is therefore (h+p, k) – [(8/3)-(3/4), -10]=(23/12, -10)