General equation of an ellipse is of the form #(x-h)^2/a^2+(y-k)^2/b^2=1#, where #(h,k)# is the center of ellipse and axis are #2a# and #2b#, with larger one as major axis an other minor axis. We can also find vertices by adding #+-a# to #h# (keeping ordinate same) and #+-b# to #k# (keeping abscissa same).
We can write the equation #9x^2+y^2+18x-6y+9=0# as
#9(x^2+2x+1)+(y^2-6y+9)=9+9-9#
or #9(x+1)^2+(y-3)^2=9#
or #(x+1)^2/1+(y-3)^2/9=1#
or #(x+1)^2/1^2+(y-3)^2/3^2=1#
Hence center of ellipse is #(-1,3)#, while major axis parallel to #y#-axis is #2xx3=6# and minor axis parallel to #x#-axis is #2xx1=2#.
Hence vertices are #(0,3)#, #(-2,3)#, #(-1,0)# and #(-1,6)#.
graph{9x^2+y^2+18x-6y+9=0 [-11.38, 8.62, -2.12, 7.88]}
