Answer
Focus: $(-\frac{25}{24},3)$
Vertex: $(-1,3)$
Work Step by Step
Recall: The parabola $(y-b)^2=4p(x-a)$ has a vertex $(a,b)$ and a focus $(a+p,b)$.
We have $6y^2+x-36y+55=0$.
Rewrite the equation above:
$6y^2-36y=-x-55$
$6y^2-36y+54=-x-55+54$
$6(y-3)^2=-(x+1)$
$(y-3)^2=-\frac{1}{6}(x-(-1))$
Then,
$(a,b)=(-1,3)$
$4p=-\frac{1}{6}\rightarrow p=-\frac{1}{24}$
$a+p=-1+(-\frac{1}{24})=-\frac{25}{24}$
So, the vertex is $(-1,3)$ and the focus is $(-\frac{25}{24},3)$.
Sketch the graph:
