Question

Identify the vertex, focus, equation of axis of symmetry, equation of directrix, direction of opening, vertex, and length of the latus rectum (focal width) of 2y² + x + 20y + 51 = 0 .

Answer 1

2y^2 + x + 20y + 51 = 0
-2(y^2+10y+25) = x+1
(y+5)^2 = -1/2 (x+1)
Now recall that the parabola
y^2 = 4px has
center at (0,0)
directrix at x = -p
focus at (p,0)
So, yours has been shifted so the vertex is at (-1,-5)
yours has p = -1/8, so
directrix x = -1 + 1/8 = -7/8
latus rectum = 4p = 1/2
directrix is the line x = -p = 1/8
it opens to the left, since the focus is at -1 - 1/8 = -9/8

Answer 2

I appreciate your answers, but at the same time, I'm sorry for not saying this, but put the answers like this:

A) Focus:

(solution here.)
---------------------------
B) Vertex:

(solution here.)
---------------------------

and so on

Answer 3

sorry - surely you can take care of the cosmetics.