Suppose we have a line $l$ and points $A$ and $B$ which are on different sides of $l$. Point $P$ is on line $l$. When we maximize $|PA-PB|$, it seems that the angle formed by $PA$ an $l$ is equal to the angle formed by $PB$ and $l$ is equal. Why is this?
Also, if we take a hyperbola with foci $A$ and $B$ and a point $P$ on the hyperbola, such that there is a line tangent to the hyperbola at point $P$, it appears that again, the angle formed by $PA$ an $l$ is equal to the angle formed by $PB$ and $l$ is equal. Again, why does this property hold for hyperbolas?
i can answer the first question.
let $A'$ be the image of $A$ on the line $l.$ look at the triangle $PBA'.$ by the triangle inequality $|PA' - PB| \le A'B$ and the equality occurs when $P, A', B$ are collinear. that shows that $AP$ and $BP$ make the same angle with $l.$