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To determine the type of conic, compute the discriminant:
\Delta = B^2 - 4AC = (-12)^2 - 4(4)(9) = 144 - 144 = 0
Since $ \Delta = 0 $, the conic is either a parabola or a degenerate conic.
To determine if it is degenerate, we compute the determinant of the full symmetric matrix of the quadratic form:
egin{bmatrix}
A & rac{B}{2} & rac{D}{2} \
rac{B}{2} & C & rac{E}{2} \
rac{D}{2} & rac{E}{2} & F
& -6 & 4 \
& 9 & -18 \