Question 763701
{{{-4x^2+9y^2-8x-54y+113=0}}}....write {{{113}}} as {{{81+36-4}}}

{{{-4x^2+9y^2-8x-54y+81+36-4=0}}}......rearrange

{{{4x^2+8x+4- 9y^2+54y-81+36=0}}}..divide by {{{-1}}} 

{{{4x^2+8x+4- 9y^2+54y-81=36}}}......group

{{{(4x^2+8x+4)- (9y^2-54y+81)= 36}}}.......divide by {{{30}}}

{{{(4x^2/36+8x/36+4/36)- (9y^2/36-54y/36+81/36)= 36/36}}}...simplify

 {{{(x^2/9+2x/9+1/9)- (y^2/4-6y/4+9/4)= 1}}}

{{{(1/9) (x^2+2x+1)-(1/4) (y^2-6y+9)= 1}}}

{{{(1/9) (x+1)^2-(1/4) (y-3)^2 = 1}}}

{{{(x+1)^2/9-(y-3)^2/4 = 1}}}

the center is at ({{{h}}}, {{{k}}}) 

 {{{h=-1}}} and {{{k=3}}}; so the center is at ({{{-1}}}, {{{3}}})


{{{a=3}}}, {{{b=2}}}, {{{c^2 = a^2+b^2}}}
{{{c^2 = 9+4}}}
{{{c^2 = 13}}}
{{{c=sqrt(13)}}} 

vertices at ({{{-4}}}, {{{3}}}) and ({{{2}}}, {{{3}}})

foci at ({{{-1-sqrt(13)}}}, {{{3}}}) and ({{{-1+sqrt(13)}}}, {{{3}}})

or ({{{-4.6}}}, {{{3}}}) and ({{{2.6}}}, {{{3}}})

assymptotes at {{{y=2x/3+11/3}}}  and {{{y=7/3-2x/3}}}


 {{{drawing( 600, 600, -10, 10, -10, 10,circle(2.6,3,0.1),locate(2.6,3,F(2.6,3)),circle(-4.6,3,0.1),locate(-4.6,3,F(-4.6,3)),circle(-1,3,0.1),locate(-1,3,C(-1,3)), graph( 600, 600, -10, 10, -10, 10,sqrt(((x+1)^2/9-1)4)+3 , -sqrt(((x+1)^2/9-1)4)+3,7/3-2x/3,2x/3+11/3 )) }}}