Question 731952
The focus given is not clear.  Either the "+" is a comma or the "-" is a comma.  No matter, if those vertices are the main vertices, then a start on the form of the equation is like, {{{(x-6)^2/36+(y+1)^2/b^2=1}}}, using center as half way between 0 and 12 on in reference to the x coordinate.


CONTINUING TO ANALYZE:
Assuming your given vertices are on the major axis and are horizontally arranged, the y coordinate of the foci MUST be in the exact same horizontal line as the major axis.  y=-1 MUST contain the foci.  If you tried to say that the coordinate pair for one of the foci is (6+11^(1/2), 11), then this is wrong.  Saying that one of the foci were (6+11^(1/2), -1) would make sense.  If this were one of the foci, then since both foci are equally distant from the center, and x=6 is one of the coordinates of the center, then the other focus must be (6-11^(1/2), -1).  
The way that a, b, and c are related is {{{a^2=b^2+c^2}}}.  The value for c is {{{6+11^(1/2)}}}, which is the focal length.  One simple algebra step gives {{{b^2=a^2-c^2}}}.


The equation for the ellipse can be amended as 
{{{(x-6)^2/36+(y+1)^2/(36-(6+11^(0.5))^2)^2=1}}},
and you can finish the arithmetic in that.

[The algebra.com system is not rendering this last equation very neatly.  Read it carefully.]