Inverse distances to points

Let $P$ and $Q$ be two distinct points on a plane. For any given point $X \neq P,Q$, define two functions $f(X)$ and $g(X)$ as follows: $$f(X) = \frac{1}{PX} + \frac{2}{QX}$$ $$g(X) = \frac{2}{PX} + \frac{1}{QX}$$ Now for any positive number $t$, let $S(t)$ be the set of all points $Y$ such that $f(Y) \leq tg(Y)$.

Prove that $S(t)$ is bounded

Prove that $S(t)$ is convex. That is, if $A,B$ are in $S(t)$ then $AB$ is in $S(t)$.

Identify all points $X$ on $S(t)$ such that $PX+QX$ is minimum.

Prove that the area of $S(t)$ is a convex function of $t$.

Powered distances

Let $p$ and $q$ be two infinitely long non-parallel lines on a plane. For any point $X$, let $d_p(X), d_q(X)$ denote distances from $X$ to $p$ and $q$ respectively. For any positive number $r$, let $S(r)$ be the set of all points $Y$ such that $d_p(Y)^{2015} + d_q(Y)^{2015} \leq r$.

Prove that $S(r)$ is bounded.

Prove that $S(r)$ is convex. That is, if $A$ and $B$ are in $S(r)$ then $AB$ is in $S(r)$.

Identify all points in $S$ such that $d_p(X)^{2014} + d_q(X)^{2014}$ is maximum.

Prove that the area of $S(r)$ is a convex function of $r$.