Trace-determinant plane: underdamped systems

[clarineterr]

Homework Statement

Describe the set of points on the trace-determinant plane that describe all underdamped systems with period 2

Homework Equations

A harmonic oscillator can be represented by the equation m$$\frac{d^{2}y}{dt^{2}}$$ +b$$\frac{dy}{dt}$$+ky=0 the Trace=-b/m and the Determinant=k/m.

For an underdamped oscillator, the period is 4m$$\pi$$/$$\sqrt{4km-b^{2}}$$
and T$$^{2}$$-4D

The Attempt at a Solution

I have been trying to solve the equation 4m$$\pi$$/$$\sqrt{4km-b^{2}}$$=2 and then find T and D, but no luck really. I think the answer is all the points above a certain parabola.

 

[kurt.physics]

I think T-4D=0 is the equation of the parabola, but I can't seem to prove it. Any help would be appreciated.