Question 975420
9x2 + 4y2 + 18x - 48y + 117 = 0 
Rewrite as
9x^2 + 18x +   ;;;; +4y^2  - 48 y  =-117;; complete first square: 9(x^2+2x+1);; second 4(y^2-12y+36)

 9(x^2+2x+1)+ 4(y^2-12y+36)=-117 + 9 +144  ;;remember to multiply the coefficient by the constant that was factored out

9(x^2+2x+1)+ 4(y^2-12y+36)=36   ;;must divide both sides by 36 to make the right side 1.

(x+1)^2/4  + (y-6)^2/9=1
Center is at (-1,6)
Major axis is along y-axis. And it is a^2=9; a=3  Vertices are at (-1,3) and (-1,9)
Minor axis is along x-axis.  And it is b^2-4; b=2  These are at (1,6) and (-3,6)
Foci are a^2-c^2=b^2  They are at sqrt (5) from center. (-1, 6+ sqrt (5)) and (-1, 6-sqrt (5))
Eccentricity is c/a which is sqrt(5)/3