In an exercise of "Rational & Nearly Rational Varieties",
I want to show a certain hypersurface contains only finite number of conics passing through a point.
The hypersurface $X$ and the point $x$ are given by $X: x_0^n-x_1x_2\cdots x_n$, $x=(1:1:\cdots:1)$.
My first attempt is to parametrize the conics as follows. $(x:y)\mapsto (a_0x^2+b_0xy+y^2:a_1x^2+b_1xy+y^2:\cdots:a_nx^2+b_nxy+y^2)$
However, after that, naive dimension counting does not give an answer, because even when the number of restrictions are bigger than the dimension of variables, they might not intersect properly.
Could anyone give me a hint to attack this problem correctly?