(x-h)^2/a^2+(y-k)^2/b^2=1 is the equation of an ellipse,
whose center is (h,k), 2a and 2b are major axis and minor axis (if a>b, but if a< b, it is reverse). In the former case major axis is parallel to x-axis and in latter case major axis is parallel to minor axis.
vertices are (h,k+-b) and (h+-a,k)
Eccentricity e can be obtained using e=sqrt(1-("minor axis")^2/("major axis")^2)
and focii are at a distance of ae on either side of center along major axis (or be if major axis is parallel to y-axis).
We can write 9x^2+25y^2-54x-50y-119=0 as
(9x^2-54x+81)+(25y^2-50y+25)=119+81+25=225
or (3x-9)^2/225+(5y-5)^2/225=1
or (x-3)^2/25+(y-1)^2/9=1 or (x-3)^2/5^2+(y+1)^2/3^2=1
center is (3,1), major axis is 2xx5=10, which is parallel to x-axis and minor axis is 2xx3=6
Vertices are (3,1+-3) and (3+-5,1) i.e. (3,4), (3,-2), (8,1) and (-2,1).
Eccentricity is e=sqrt(1-(3/5)^2)=sqrt(9/25)=4/5 and ae=4
and focii are (3+-4,1) i.e. (-1,1)and (7,1).
They appear as follows:
graph{(9x^2+25y^2-54x-50y-119)((x-3)^2+(y-1)^2-0.03)((x-3)^2+(y-4)^2-0.03)((x-3)^2+(y+2)^2-0.03)((x-8)^2+(y-1)^2-0.03)((x+2)^2+(y-1)^2-0.03)((x-7)^2+(y-1)^2-0.03)((x+1)^2+(y-1)^2-0.03)=0 [-7.14, 12.86, -3.78, 6.22]}
