SOLUTION: Is {{{3y^2 - 16y - x^2 +36 = 0}}}, the conic section is a hyperbola; a circle

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Question 1015227: Is 3y%5E2+-+16y+-+x%5E2+%2B36+=+0, the conic section is a hyperbola; a circle
Found 2 solutions by ikleyn, robertb:
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
Is 3y^2 - 16y - x^2 + 36 = 0 a circle?
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No.


Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Let Ax%5E2+%2BBxy+%2BCy%5E2+%2B+Dx+%2B+Ey+%2BF+=+0 be the general form of the equation of a conic section in cartesian coordinates.
Then
if B%5E2+-+4AC+%3E+0, the conic section is a hyperbola;
if B%5E2+-+4AC+=0, the conic section is a parabola;
if B%5E2+-+4AC+%3C+0, the conic section is a ellipse.
The given equation equation can be rewritten as

x%5E2+-3y%5E2+%2B16y-36+=+0 in general form
Since for the given equation B%5E2+-+4AC+=+0+-+4%281%29%28-3%29+=+12%3E0, the conic section is a hyperbola. The equation does NOT represent a circle.