Find the foci of the ellipse 6x^{2}+2y^{2}-54x-10y=0

6x^{2}+2y^{2}-54x-10y=0

(6x^{2}-54x)+(2y^{2}-10y)=0

6(x^{2}-9x)+2(y^{2}-5y)=0

6(x^{2}-9x+81/4)+2(y^{2}-5y+25/4)=6(81/4)+2(25/4)

6(x-9/2)^{2}+2(y-5/2)^{2}=243/2+25/2=134

The major axis is vertical (a^{2}>b^{2}
always and therefore 67=a^{2} and 67/3=b^{2}) where h=9/2,
k=5/2, a=√67, b=√67/3.

c^{2}=a^{2}-b^{2}

c=√(a^{2}-b^{2})=√67-67/3=√(134/3)

The foci are therefore at (9/2, 5/2+√(134/3)) and (9/2, 5/2-√(134/3))