The green line below is tangent drawn at point $P$. This construction uses the fact that the angle $PFT$ is $90$ degrees. But it doesn't give any explanation of why it must be $90$. Is there any simple way to see this with or without calculus?
Intuitively, when $P$ are right above $F$, it is clear that the angle is 90 because $PF$ is vertical and $FT$ is horizontal. As $P$ moves to the right, it seems $T$ also moves upkeep the angle at focus $90$. I'm not that sure how to approach proving things like these... Help appreciated.