Question

is 2x^2-5y^2+4x-6=0
a hyperbola or an ellipse?

Answer 1

It is a hyperbola because the coefficients of the x^2 and y^2 terms have opposite signs.

For arbitrarily large x, whether positive or negative, where are always two values of y that satisfy the equation.

For an ellipse, the signs of the x^2 and y^2 terms are the same.