Question 854012
Identify the vertex, focus and directrix of the parabola with the equation x^2-6x-8y+49=0
***
x^2-6x-8y+49=0
x^2-6x-8y=-49
complete the square:
(x^2-6x+9)-8y=-49+9
(x-3)^2=8y-40
(x-3)^2=8(y-5)
This is an equation of a parabola that opens up:
Its basic equation: (x-h)^2=4p(y-k), (h,k)=coordinates of vertex
For given problem:
vertex:(3,5)
axis of symmetry: x=3
4p=8
p=2
focus: (3,7)(p-distance above vertex on the axis of symmetry)
directrix: y=3(p-distance below vertex on the axis of symmetry)

see graph below as a visual check:

{{{ graph( 300, 300, -10, 10, -10, 10,(x^2-6x+49)/8) }}}