Question 746555
  {{{18x^2+4y^2-108x+16y=106 }}}

  {{{(18x^2-108x+__)+(4y^2+16y+___)-106 =0}}}....complete the square

{{{(18x^2-108x+162)-162+(4y^2+32y+16)-16-106=0}}}

{{{(18x^2-108x+162)+(4y^2+32y+16)-284=0}}}.......factor out {{{18}}} from first group and {{{4}}} from second group

{{{18(x^2-6x+9)+4(y^2+8y+4)-284=0}}}


{{{18(x-3)^2+4(y+2)^2=284}}}


{{{18(x-3)^2/284+4(y+2)^2/284=284/284}}}


{{{(x-3)^2/15.78+(y+2)^2/71=1}}}...round {{{15.78}}} to whole number


{{{(x-3)^2/16+(y+2)^2/71=1}}}....=> {{{h=3}}} and {{{k=-2}}}


the center is at: ({{{3}}},{{{-2}}})

 vertices: since the {{{y}}} term has the larger denominator, the ellipse is {{{vertical}}} and
vertices ({{{3}}}, {{{-2+-sqrt(71)}}} ≅ ({{{3}}}, {{{-6.43}}}) and ({{{3}}},{{{- 10.43}}}) 

and foci:
({{{3}}}, {{{-2+-sqrt(497)/3}}} ≅ ({{{3}}}, {{{-2+-7.43}}} ≅

({{{3}}}, {{{-5.43}}}) and ({{{3}}},{{{- 9.43}}}) 


{{{drawing( 600, 600, -15, 15, -15, 15,circle(3,-2,0.2), graph( 600, 600, -15, 15, -15, 15, -sqrt(-4.5(x-3)^2+71)-2,sqrt(-4.5(x-3)^2+71)-2)) }}}