I have the following problem and seems to stuck with some basic understanding of irreducible and/or non-singular varieties.
In $\mathbb{P}^3$ we have an irreducible variety $A$ given by two equations $tx=p$ and $ty=q$ where $p$ and $q$ are quadratic forms in $x$, $y$ and $z$. We need to argue that the map $\phi(x,y,z,t)\to(x,y,z)$ restricted to $A$ is an isomorphism of $A$ to the plane variety given by $yp=xq$.
The problem I have is the following. So, it is straightforward to see that $\phi$ restricted to $A$ maps to $D$, the set of solutions of the plane equation. We can further define the inverse map by $(x,y,z)\to(x^2,xy,xz,p)$ for $x\neq0$, and $(x,y,z)\to(xy,y^2,yz,q)$ for $y\neq0$. This gives us the inverse rational map for all $x \neq 0 \neq y$. What I want is to further argue that $(0,0,1)$ is not in the image of $\phi$ (otherwise, there is a whole subspace mapped to this point as argued below).
The way I try to think about it is that if $(0,0,1)$ is in the image, then at this point both quadratic forms $p$ and $q$ are $0$, and, hence, every point $(0,0,1,t)$ is in $A$. This means, in particular, that both $p$ and $q$ are missing the term $z^2$. Now, I do not have enough vision of irreducibility to conclude this argument. It seems that I miss some basic understanding which is unfortunately not provided by a book I read.
I have also checked when the intersection of two quadratic forms is irreducible. I found a result where it says that if the two quadratic forms are given by matrices $A$ and $B$, then their intersection is irreducible iff $\det(A-\lambda B)=0$ has different roots. Assuming this is true, and assuming that $p$ and $q$ are both missing the term $z^2$ (as well as $t^2$), taking $A$ and $B$ for quadratic forms $p-tx$ and $q-ty$, indeed, we have the determinant $l(\lambda)^2$ where $l(\lambda)$ is a linear function in $\lambda$. This means that the determinant equals zero at a single point $\lambda$. This seems to support my idea, but the problem is that this result is not a part of what I want to use (it is from a paper I found), so I hope there is some way to argue what I want without using any additional results, but on an intuitive level. For example, by showing explicitly a decomposition of $A$ in this case.