Use complete the square and identify the conic.
X^2+y^2+4x-6y+1=0
A. (X-2)^2+(y+3)^2=2 square root 3
B. (X+2)^2+(y-3)^2=12
To complete the square for this equation, we need to rewrite it as:
(X^2 + 4x) + (y^2 - 6y) + 1 = 0
(X^2 + 4x + 4) - 4 + (y^2 - 6y + 9) - 9 + 1 = 0
(X + 2)^2 - 4 + (y - 3)^2 - 9 + 1 = 0
(X + 2)^2 + (y - 3)^2 - 12 = 0
As we can see, by completing the square, we get the conic in the form (X + h)^2 + (y + k)^2 = r^2, which represents a circle.
Therefore, the correct answer is B. (X + 2)^2 + (y - 3)^2 = 12.