Which equation describes a circle?
A: -2y2+y+2x2-5x-3=0
B: -3x2+x-6y2+y=4
C: -3y2+x+y=3
Identify the conic section with the given equation 5x2-6y2-9x+2y+3=0
p.s. the twos are supposed to be raised to that power
None of those is a circle. In a general quadratic form, if the coefficients of x^2 and y^2 are equal, it's a circle; unequal but the same sign, an ellipse; one zero and one non-zero, a parabola; of different sign, a hyperbola (rectagular if the coefficients are equal in absolute value). So you have a rectangular hyperbola, an ellipse and a parabola respectively.
The quadradic in form of,
$\displaystyle
ax^2+2hxy+by^2+2gx+2fy+c=0$
can de determined by looking a the relationship between,
$\displaystyle h^2$ and $\displaystyle ab$.
$\displaystyle h^2=ab$---->parabola
$\displaystyle h^2>ab$---->hyperbola
$\displaystyle h^2<ab$---->ellipse
In your case we have #2 which is a hyperbola.
[Note: Some cases where not considered such as the coditions for a general quadradic to represent a line or to represent a point]