I'm trying to find the length of $PQ$ but the best thing I have done so far is finding that the point $T$ is $(0,4)$, as well as finding the distance between the two turning points to be $6$.
Can anyone give me a hint at how to derive an expression for $PQ$?
If $TP = x$, then $Q=(x,10-x^2)$ and so $PQ= 10-x^2-4=6-x^2$.