Question 113835
the general equation for this ellipse is {{{(((x-h)^2)/b^2)+(((y-k)^2)/a^2)=1}}}


this is an ellipse with a semimajor axis a and semiminor axis b, centered at the point (h,k)


if c is the distance from the center to a focus, then c^2=a^2-b^2


{{{4(x^2+4x)+y^2+2y+1=0}}} ___ {{{4(x^2+4x+4)+y^2+2y+1=16}}} ___ {{{4((x+2)^2)+(y+1)^2=16}}}


dividing by 16 gives {{{(((x+2)^2)/4)+(((y+1)^2)/16)=1}}}


the center (-2,-1) is midway between the foci
___ a=4; is the distance from center to vertex
___ b=2, so c^2=(4^2-2^2) ___ c^2=12


vertices ___ (-2,3) and (-2,-5)
minor axis endpoints ___ (-4,-1)(0,-1)
foci ___ (-2,(sqrt(12)-1)) and (-2,-((sqrt(12))+1))