Question 1121763


{{{9x^2 + 4y^2 - 24y -72  + 144 = 0 }}}...-> I guess you got {{{72x}}}

{{{9x^2 + 4y^2 - 24y -72x + 144 = 0 }}}

{{{(9x^2-72x )+ (4y^2 - 24y ) + 144 = 0 }}}

{{{9(x^2-8x )+4 (y^2 - 6y ) + 144 = 0 }}}

{{{9(x^2-8x+b^2 ) -9b^2+4 (y^2 - 6y +b^2) -4b^2+ 144 = 0 }}}

{{{9(x^2-8x+4^2 ) -9*4^2+4 (y^2 - 6y +3^2) -4*3^2+ 144 = 0 }}}

{{{9(x-4)^2  -9*16+4 (y - 3)^2 -4*9+ 144 = 0 }}}

{{{9(x-4)^2  -144+4 (y - 3)^2 -36+ 144 = 0 }}}

{{{9(x-4)^2 +4 (y - 3)^2 -36 = 0 }}}

{{{9(x-4)^2 +4 (y - 3)^2 = 36 }}}

{{{9(x-4)^2/ 36 +4 (y - 3)^2 /36 = 36 /36 }}}

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

major axis is vertical, so compare to  {{{(x-h)^2/b^2+(y-k)/a^2 =1}}}, you see that 
{{{h=4}}}
{{{k=3}}}

{{{b^2=4}}}=>{{{b=2}}}

{{{a^2=9}}}=>{{{a=3}}}

eccentricity : {{{c/a}}}

first find {{{c}}}:

{{{c^2=a^2-b^2}}}
{{{c^2=3^2-2^2}}}
{{{c^2=9-4}}}
{{{c^2=5}}}
{{{c=sqrt(5)}}}


 eccentricity: {{{sqrt(5)/3}}} ≈ {{{0.76}}}

center: ({{{4}}}, {{{3}}})
foci : ({{{4}}}, {{{3 - sqrt(5)}}}) and ({{{4}}}, {{{3 + sqrt(5)}}}) 

or ≈ ({{{4}}}, {{{0.76}}})  and  ({{{4}}}, {{{5.24}}})

vertices:({{{4}}}, {{{0}}})  and  ({{{4}}},{{{ 6}}})

covertices: ({{{6}}}, {{{3}}})  and  ({{{2}}}, {{{3}}})