For real numbers $t \neq 0,$ the point
\[(x,y) = \left( t + \frac{1}{t}, t - \frac{1}{t} \right)\]is plotted.  All the plotted points lie on what kind of curve?

(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola

Enter the letter of the correct option.
Note that
\[x^2 - y^2 = \left( t + \frac{1}{t} \right)^2 - \left( t - \frac{1}{t} \right)^2 = \left( t^2 + 2 + \frac{1}{t^2} \right) - \left( t^2 - 2 + \frac{1}{t^2} \right) = 4,\]so
\[\frac{x^2}{4} - \frac{y^2}{4} = 1.\]Thus, all the plotted points lie on a hyperbola.  The answer is $\boxed{\text{(E)}}.$