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In class, I've defined a projective conic on $\mathbb{P}(V)$ as an element of $\mathbb{P}(Q(V))$, where $V$ is a $K$-vector space and $Q(V)$ is the field of quadratic forms on $V$. I need to prove the following:

Given two points $A,B$ of the complex or real projective plane, the pencils $\mathcal{H}_A,\mathcal{H}_B$ of lines at $A$ and $B$ respectively, and a projectivity $\psi:\mathcal{H}_A\to\mathcal{H}_B$, the set of points $r\ \cap \psi (r)$ are all contained in a conic.

So, what I need is to prove that both definitions are equivalent (for a $2$-dimensional projective space).

EDIT: I'm thinking of using the immersion of the affine into the projective plane, and hopefully be able to define an affine conic that will give me the projective conic I am looking for.

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