Question 856394
{{{4x^2+y^2+24x-10y+45=0}}}
{{{4x^2+24x+y^2-10y=-45}}}
{{{4x^2+24x+36+y^2-10y+25=-45+36+25}}}
{{{4(x^2+6x+9)+(y-5)^2=16}}}
{{{4(x+3)^2+(y-5)^2=16}}}
{{{4(x+3)^2/16+(y-5)^2/16=1}}}
{{{(x+3)^2/4+(y-5)^2/16=1}}}
{{{(x+3)^2/2^2+(y-5)^2/4^2=1}}}
That is the equation of an ellipse
centered at {{{highlight("( -3 , 5 )")}}} ,
with major axis parallel to the y-axis
and semi-major axis {{{a=4}}} ,
with minor axis parallel to the x-axis
and semi-major axis {{{b=2}}} ,
and a focal distance {{{c}}} such that
{{{b^2+c^2=a^2}}} so
{{{2^2+c^2=4^2}}} or {{{4+c^2=16}}} so
{{{c^2=16-4}}}-->{{{c^2=12}}}-->{{{c=sqrt(12)}}}-->{{{c=2sqrt(3)}}}--> {{{c=about3.464}}}
{{{drawing(300,300,-7,3,-1,10,
grid(1),red(circle(-3,5,0.1)),
red(arc(-3,5,4,8,0,360)),
red(circle(-3,9,0.1)),
red(circle(-3,1,0.1)),
green(circle(-3,8.464,0.1)),
green(circle(-3,1.536,0.1)),
locate(-6.5,9.6,red(vertex)),
locate(-6.5,0.9,red(vertex)),
locate(-4.6,5.5,red(center)),
locate(-4.6,1.9,green(focus)),
locate(-4.6,8.9,green(focus)),
red(arrow(-4.8,9.3,-3.15,9.1)),
red(arrow(-4.8,0.6,-3.15,0.9))
)}}}
The vertices are 4 units above and 4 units below the center,
at {{{highlight("( -3 , 9 )")}}} and {{{highlight("( -3 , 1 )")}}} .
The foci are {{{2sqrt(3)}}} above and {{{2sqrt(3)}}} below the center,
at {{{highlight("( -3 , 5+2 sqrt(3))")}}} and {{{highlight("( -3 , 5-2 sqrt(3))")}}} .
The length of the major axis is the distance between the vertices
{{{2a=2*4=highlight(8)}}} .
The length of the minor axis is
{{{2b=2*2=highlight(4)}}} .