Question 1043332
{{{9x^2+16y^2-126x+64y=71}}}
{{{9x^2-126x+16y^2+64y=71}}}
{{{9(x^2-14x)+16(y^2+4y)=71}}}
Thge expressions in brackets remindme of some squares:
{{{x^2-14x+49=(x-7)^2}}} , so {{{9(x-7)^2)=9(x^2-14x+49)=9x^2-126x+441}}} , and
{{{y^2+4y+4=(y+2)^2}}} , so {{{16(y+2)^2)=16(y^2+4y+4)=16y^2+64y+64}}} .
Adding {{{441+64}}} to both sides of the equal sign in the  original equation, we have
{{{9x^2+16y^2-126x+64y+441+64=71+441+64}}}
{{{9x^2-126x+441+16y^2+64y+441+64=576}}}
{{{9(x^2-14x+49)+16(y^2+4y+4)=576}}}
{{{9(x-7)^2+16(y+2)^2=576}}}
Dividing both sides of the equal sign in the equation above by {{{9*16}}} , we have
{{{(x-7)^2/16+(y+2)^2/9=576/(9*16)}}}
{{{(x-7)^2/16+(y+2)^2/9=4}}}
Dividing both sides of the equal sign in the equation above by {{{4}}} , we have
{{{(x-7)^2/64+(y+2)^2/36=1}}} , or {{{(x-7)^2/8^2+(y+2)^2/6^1=1}}} .
That is the equation of an ellipse centered at {{{"( 7 , -2 )"}}} ,
the point with {{{highlight(system(x=7,y=-2))}}} .
The semi-major axis length is {{{a=8}}} , and
semi-minor axis length is {{{b=6}}} .
The {{{a^2=8^2=64}}} is dividing the term with {{{x}}},
so the major axis is parallel to the x-axis,
and since the center has {{{y=-2}}},
the major axis is the line {{{y=-2}}} .
On that major axis, are the vertices and foci.
The minor axis is parallel to the y-axis,
and since the center has {{{x=7}}},
the major axis is the line {{{x=7}}} .
We know that the focal distance of an ellipse, {{{c}}} , can be found as 
{{{c=sqrt(a^2-b^2)}}} , so in this case
{{{c=sqrt(64-36)=sqrt(28)=4sqrt(7)}}} . An approximate value is {{{c=5.29}}} .
The locations of the foci are on the major axis,
at a distance {{{c=4sqrt(7)=about5.29}}} to either side of the center,
so one the coordinates of the foci are
{{{highlight(system(x=7-4sqrt(7),y=-2))}}} , or about {{{"( 1.71 , -2 )"}}} for one focus,
and {{{highlight(system(x=7+4sqrt(7),y=-2))}}} , or about {{{"( 12.29 , -2 )"}}} for the other focus.
Similarly, the vertices are on the major axis,
at a distance {{{a=8}}} to either side of the center,
so one the coordinates of the vertices are
{{{system(x=7-8,y=-2)=highlight(system(x=-1,y=-2))}}} ,
and {{{system(x=7+8,y=-2)=highlight(system(x=15,y=-2))}}} .
The covertices are on the minor axis,
at a distance {{{b=6}}} to either side of the center,
so one the coordinates of the vertices are
{{{system(x=7,y=-2-6)=highlight(system(x=7,y=-8))}}} ,
and {{{system(x=7,y=-2+6)=highlight(system(x=7,y=4))}}} .
The ellipse with axes and foci looks like this:
{{{drawing(500,350,-3,17,-9,5,
grid(0),
green(arrow(-6,-2,20,-2)),
green(arrow(7,-10,7,10)),
arc(7,-2,16,12,0,360),
green(circle(7-sqrt(28),-2,0.1)),
green(circle(7+sqrt(28),-2,0.1))
)}}}