Suppose that $X$ is a quasi-projective $k$-scheme with a right $B$-action (where $B$ is a linear algebraic group or Lie group) and that the quotient $X/B$ exists. Let the canonical projection $X \to X/B$ be a principal $B$-bundle (with fibre $B$).
Now consider the associated bundle $f: Y \to X/B$ with fibre $\mathbb{P}^1$. Suppose that $X/B$ is a projective scheme. Can someone please help me prove the following:
(1) $f$ is a projective morphism.
(2) If $X/B$ is a projective scheme, then so is $Y$.