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### Hyperbolic Functions

This applet shows the graph of the parabola $y=\dfrac{x^2}{4a}$ with its focus $F$ at $(0,a)$. The value of $a$ can be changed by sliding the left slider.
The parabola can be rolled along the $x$-axis by moving the right slider.
As we roll the parabola in this way, what is the (cartesian equation of the) locus of the focus $F$?