Question

The curve x^4+y^4=4a^2xy is symmetric with respect to the origin.
True or false with full explanation

Answer 1

Let $(x_0, y_0)$ is the point of the equation, that is $x_0^4+y_0^4=4a^2x_0y_0.$

Consider $(-x_0, -y_0)$ – symmetric point with respect to the origin.

Substitute it into the equation:

$(-x_0)^4+(-y_0)^4=4a^2(-x_0)(-y_0),$

or $x_0^4+y_0^4=4a^2x_0y_0.$

So the curve is symmetric with respect to the origin.