Calculus
# Parametric Equations

$\begin{cases} x = e^t \\ y = e^{2t} - 1 \end{cases}$

What is the shape of the curve described by the above parametric equation?

Two clowns, Twinkle and Jingle, are throwing pies at each other. Twinkle throws a pie toward Jingle from $500$ centimeters away. Its flight path is given by parametric equations $\begin{cases} x &=& 100t \\ y &=& 80t - 16t^2 \end{cases}$ where $t$ is time in seconds.

Two seconds later Jingle launches an interceptor pie from his location with the flight path $\begin{cases} x &=& 500 - 500(t-2) \\ y &=& K(t-2) - 16(t-2)^2 \end{cases}$ Find the value of $K$ which will guarantee that the interceptor pie will hit its target (the pie thrown by Twinkle).