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.
Here's an image that can help you visualize the reflective property (incoming horizontal rays are reflected toward the focus) and why the tangent bisects $\angle FPQ$.