Final answer:
The equation provided does not represent an ellipse in its standard form, so it's not possible to state definitively which lines the axes of an ellipse are on without further transformation. If we could manipulate the given equation into an ellipse's standard form, the likely candidates for axes would be the lines y=x and x=-y.
Step-by-step explanation:
The equation x²-xy+y²=4 does not represent a standard form of an ellipse. However, you might be trying to determine which lines the axes of an ellipse would be on if the equation could be transformed into that of an ellipse. An ellipse is a closed curve wherein the distances from the two foci to any point on the curve are equal, and it is symmetric with respect to its major and minor axes.
Typically, the axes of an ellipse are perpendicular to each other. The lines given as options (y=x, x=-y) are also perpendicular to each other and could potentially serve as axes for an ellipse if the equation could be manipulated into a standard form. However, without proper transformation, it is impossible to definitively state along which lines the axes of the ellipse lie based on the provided equation.
If the given equation was indeed an ellipse (which would be when rearranged into the form αx²+βy²=1), the most likely candidates for its axes would be the diagonal lines y=x and x=-y, as option 3) x=y and option 4) x=-y, since they define directions that are mutually orthogonal.