Find all local extremes of a function with parameter.

Question

$$f(x,y)=xy+e^{a(x^2+y^2)}$$ I calculated all the first- and second-order partial derivatives: $$ \frac{\partial f}{\partial x} = y+2axe^{a(x^2+y^2)}\\ \frac{\partial f}{\partial y} = x+2aye^{a(x^2+y^2)}\\ \frac{\partial^2 f}{\partial^2 x} = 2ae^{a(x^2+y^2)}(1+2ax^2)\\ \frac{\partial^2 f}{\partial^2 y} = 2ae^{a(x^2+y^2)}(1+2ay^2)\\ \frac{\partial^2 f}{\partial x\partial y} = 1+4a^2xye^{a(x^2+y^2)} $$ But checking if the Hessian matrix is definite for every case is really tedious. Is there another way?