jmc

We try to rewrite the given equation in a standard form for a conic section. Expanding the left-hand side, we get $$x^2-y^2=-2y^2+1.$$Then adding $2y^2$ to both sides gives $$x^2+y^2=1,$$which is the equation for a $\boxed{\text{(C)}}$ circle (in fact, the unit circle centered at the origin).