Question 1004568
{{{x^2+4y^2+6x+16y+21=0}}}

{{{(x^2+6x)+(4y^2+16y)+21=0}}}...........complete square

{{{(x^2+6x+b^2)-b^2+(4y^2+16y+b^2)-b^2+21=0}}}

{{{(x^2+6x+b^2)-b^2+4(y^2+4y+b^2)-4b^2+21=0}}}

for {{{x}}} part we have {{{a=1}}} and {{{2ab=6}}}=>{{{2*1*b=6}}}=>{{{b=3}}}

for {{{y}}} part we have {{{a=1}}} and {{{2ab=4}}}=>{{{2*1*b=4}}}=>{{{b=2}}}


{{{(x^2+6x+3^2)-3^2+(4y^2+16y+2^2)-4*2^2+21=0}}}

{{{(x+3)^2-9+4(y+2)^2-16+21=0}}}

{{{(x+3)^2+4(y+2)^2-25+21=0}}}

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

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


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

{{{(x+3)^2/4+(y+2)^2/1=1}}}=>{{{h=-3}}},{{{k=-2}}}, {{{a=2}}},{{{b=1}}}

{{{c^2=a^2-b^2}}}
{{{c^2=2^2-1^2}}}
{{{c^2=4-1}}}
{{{c^2=3}}}
{{{c=sqrt(3)}}}


the center:({{{-3}}},{{{-2}}} )
semimajor axis length: {{{ 2}}}
semiminor axis length: {{{1}}}

foci: ({{{h-c, -2}}}) and ({{{h+c, -2}}})

({{{-3-sqrt(3)}}}, {{{-2}}}) and  ({{{-3+sqrt(3)}}}, {{{-2}}})

or ({{{-4.7}}},{{{ -2}}}) and  ({{{-1.3}}},{{{ -2}}})

vertices: ({{{h-a}}}, {{{-2}}}) and ({{{h+a}}}, {{{-2}}})
({{{-5}}}, {{{-2}}})  and  ({{{-1}}}, {{{-2}}})

{{{ graph( 600, 600, -6, 6, -6, 6, sqrt((4-(x+3)^2)/4)-2,-sqrt((4-(x+3)^2)/4)-2) }}}