Rewritting QCQP as SOCP

Question

I have a convex function with convex constraints. $$f(p) = \min_\limits{\| b_{0} - A_{0} x \|_{2} \le p} \| x \|_{2} = \min_\limits{\| b_{0} - A_{0} x \|_{2} = p} \| x \|_{2}$$ I was researching a lot about such problem, and found interesting question right here. It loots like it is pretty simple to rewrite my task to SOCP in following format, if I am right. $$\min \limits_{~~~~~~||x||_{2} \le y \\~~||A_{0}x - b_{2}||_{2} \le p} y $$ However I don't understand what is happening here. We are minimizing y, so it is taken from real numbers, but what are $x$ here then? Is it some fixed vector or what? It should not be, but what is it then. I don't get it completly. Rewritting this problem as SOAP would allow me to use this method to compute that function.