Hi! I really need help with my homework, but I don't have anybody to ask. The problem asks me to find the vertex, focus, and directrix of the parabola: x^2-12x-48y+276=0. All help is apprectiated! Thank you.
writing it in terms of y first of all
48y = x^2 - 12x + 276 or
48y = x^2 -12x + 36 - 36 + 276
48y = (x - 6)^2 + 240
y = (1/48) (x-6)^2 + 5
so the vertex is (6,5)
the x of the focus is the same as that for the vertex
so x = 6
for y = (1/48)x^2 - (1/4)x + 23/4
the y of the focus is c - (b^2 - 1)/(4a)
= 23/4 - (1/16 - 1)/(1/12)
= 23/4 + (15/16)(12)
= 17
So the focus is (6,17)
directrix is Y = c - (b^2 + 1)/(4a)
= 23/4 - (1/16 + 1)/(1/12)
= 23/4 - (17/16)(12)
= 23/4 - 51/4
= -7
vertex (6,5)
focus ( 6,17)
directrix : y = -7