The simplest way to find a parametrization of the plane projective curve $XZ-Y^2=0$. I have to explain to some first year math students that the projective algebraic set $\textbf{Z}(XZ-Y^2)\subset\mathbb P^2_k$ is 
$$V=\{(a^2_0:a_0a_1:a^2_1)\subset\mathbb P^2_k \,:\, \textrm{for}\; a_0,a_1\in k\}$$
It is evident that $V\subseteq \textbf{Z}(XZ-Y^2)$, but which is, in your opinion, the simplest way to show the other inclusion? 

Just a note, lots of number theory of quadratic forms happens because all null vectors of your form, with rational integer entries, are integer multiples of
$$ \left( u^2 , u v, v^2  \right) $$
with $\gcd(u,v)=1.$
I'm just sayin'
