$V(x^2+y^2+z^2)$ and $V(xz−y^2,yw−z^2,xw−yz)$ are isomorphic.

Question

I saw a statement that read:

$$X:=V(x^2+y^2+z^2)\subset \mathbb{P}^2$$ and $$Y :=V(xz−y^2,yw−z^2,xw−yz)\subset \mathbb{P}^3$$

are isomorphic.

I don't immediately see why this is true. Thoughts?