SOLUTION: Find the vertices, the endpoints of the minor axis, and the foci of the ellipse {{{4x^2+16x+y^2+2y+1=0}}}. I dont even know where to begin on this problem, I would really apprec

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Find the vertices, the endpoints of the minor axis, and the foci of the ellipse {{{4x^2+16x+y^2+2y+1=0}}}. I dont even know where to begin on this problem, I would really apprec      Log On


   

Question 113835: Find the vertices, the endpoints of the minor axis, and the foci of the ellipse 4x%5E2%2B16x%2By%5E2%2B2y%2B1=0.
I dont even know where to begin on this problem, I would really appreciate it if someone could help me. Thank you!!!
Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
the general equation for this ellipse is %28%28%28x-h%29%5E2%29%2Fb%5E2%29%2B%28%28%28y-k%29%5E2%29%2Fa%5E2%29=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%28x%5E2%2B4x%29%2By%5E2%2B2y%2B1=0 ___ 4%28x%5E2%2B4x%2B4%29%2By%5E2%2B2y%2B1=16 ___ 4%28%28x%2B2%29%5E2%29%2B%28y%2B1%29%5E2=16

dividing by 16 gives %28%28%28x%2B2%29%5E2%29%2F4%29%2B%28%28%28y%2B1%29%5E2%29%2F16%29=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))