Step 1
A nondegenerate conic section of the form,
\(Ax^2 + By^2 + Cx + Dy + E=0\)
Where A and B both are not zero.
1) A circle if \(A = B\)
2) A parabola if \(AB = 0\)
3) An ellipse if A is not equal to \(B and AB > 0\)
4) A hyperbola if \(AB < 0\)
Step 2
Given that,
\(3x^2 + 2y^2 + 12x − 4y + 2 = 0\)
Here \(A = 3 and B = 2\)
So A and B are not equal and \(AB = (3) (2) = 6 > 0\)
Hence the graph of the equation is ellipse.