F and Cauchy from independent normals

Let $X_{1},X_{2},X_{3}$ be three independent random variables such that $X_{1}\sim N\bigl(0,\frac{1}{2}\bigr)$, $X_{2}\sim N(0,2)$, and $X_{3}\sim N(0,4)$. Consider the following statements. $P$: $\dfrac{16X_{1}^{2}}{2X_{2}^{2}+X_{3}^{2}}$ has $F$ distribution with $1$ and $2$ degrees of freedom. $Q$: $\dfrac{2X_{1}-X_{2}}{2X_{1}+X_{2}}$ has standard Cauchy distribution. Then