SOLUTION: Discuss the conic 4x2-y2+8x+2y-1=0 find its foci, vertices and asymptotes if any

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Question 971232: Discuss the conic 4x2-y2+8x+2y-1=0
find its foci, vertices and asymptotes if any
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
4x%5E2-y%5E2%2B8x%2B2y-1=0.......group x's and y's
%284x%5E2%2B8x%29-%28y%5E2-2y-1%29=0.........complete squares
4%28x%5E2%2B2x%29-%28y-1%29%5E2=0
4%28x%5E2%2B2x%2B1%5E2%29-4%2A1%5E2-%28y-1%29%5E2=0
4%28x%2B1%29%5E2-4-%28y-1%29%5E2=0
4%28x%2B1%29%5E2-%28y-1%29%5E2=4..........divide by 4
4%28x%2B1%29%5E2%2F4-%28y-1%29%5E2%2F4=4%2F4
%28x%2B1%29%5E2-+%28y-1%29%5E2%2F4+=+1 ..........you have a hyperbola
compare it to the standard formula %28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2+=+1
as you can see, h=-1, k=1, a=1, and b=2
then c=sqrt%281%5E2%2B2%5E2%29=sqrt%285%29
so,
the center is at (-1,1
foci:
((h-c, a) and (h%2Bc,+a))
((-1-sqrt%285%29, 1) and (-1%2Bsqrt%285%29, 1))
or, approximately ((-3.24,+1) and (1.24,+1))
vertices: | (-2, 1) and (0, 1)
semi-major axis length: 1
semi-minor axis length: 2
asymptotes: y+=+2x%2B3 and y+=+-2x-1