Is the following relation reflexive, symmetric and transitive?
The relation is symmetric, because if $y^2 \le 9-x^2$ then we can write $$y^2 \le 9-x^2 \Leftrightarrow y^2+x^2 \le 9 \Leftrightarrow x^2 \le 9-y^2$$
For the transitivity, we can use again that $x \sim y \Leftrightarrow x^2+y^2 \le 9$ to construct a counterexample: $x=3, y=0, z=3$. We have $x \sim y$ and $y \sim z$ but $x \not \sim z$.