Final answer:
For the given equation to represent a circle, it requires that the coefficients of x² and y² be equal (a = b) and that the xy term's coefficient (h) must be zero.
Step-by-step explanation:
The equation ax²+2hxy+by²+2gx+2fy+c=0 can represent a circle under a specific condition. This condition is derived from the general equation of a circle, which can be compared to the equation of a conic section. For the equation given to represent a circle, it must satisfy the condition that a = b and h = 0. This implies that the coefficients of x² and y² are equal and the xy term must be absent, as a circle's radius is constant in all directions and it does not have any cross-product terms. When a = b and h = 0, the resulting equation simplifies to ax²+ay²+2gx+2fy+c=0, which can be rewritten in standard form to represent a circle.