Find the vertex, focus, and directrix of the parabola

y^2 – 8y – 32 – 12x = 0

... y² - 8y - 32 - 12x = 0

∴ y² - 2(4)y = 12x + 32

∴ y² - 2(4)y + 4² = 12x + 32 + 4²

∴ (y-4)² = 12(x+4)

∴ Y² = 4aX, ............ X=x+4, Y=y-4, 4a=12 ......... (1)
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∴ Vertex ≡ ( X=0, Y=0 ) ≡ ( x+4=0, y-4=0 ) ≡ (-4,4) .......... Ans.(i)

∴ Focus ≡ ( X=a, Y=0 ) ≡ ( x+4=3, y-4=0 ) ≡ (-1,4) .......... Ans.(ii)

∴ directrix : X = -a, ... x+4 = -3, ... x = -7 ........................... Ans.(iii)
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