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Let $X\subset\mathbb{P}^{N}$ be a $n$-dimensional algebraic variety and let $x\in X$. Let us suppose that $$ \hat{Y}=\{\text{quadrics $Q\subset X$ of dimension $\frac{n}{2}$ such that $x\in Q$}\} $$ is parametrized by a quadric hypersurface $Y$ contained in certain projective space $\mathbb{P}^{\frac{n}{2}+1}$. I would like to prove that if $Q_{1},Q_{2},Q\in \hat{Y}$ correspond to $y_{1},y_{2},y\in Y$ respectively, and $y\in\langle y_{1},y_{2}\rangle\subset Y$, then $$ \dim (Q_{i}\cap Q)>0\text{ }\forall i\in\{1,2\} \Leftrightarrow y\in T_{y_{1}}Y\cap T_{y_{2}}Y. $$ Feel free to add any reasonable hypothesis in order to the result to hold (For example, does the assumption that $Q_{i}\cap Q$ is a linear subspace help?).

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