Let $p_1,...,p_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p_1,...,p_8$. I am interested in what happens for nodal rational curves of degree $4$.

Now, let us suppose we have seven points $p_1,...,p_7\in\mathbb{P}^{3}$ in linear general position.

**How many irreducible, rational curves of degree $4$ pass through $p_2,...,p_7$ and have a singular point of multiplicity $2$ at $p_1$ ?**

Such a curve can be constructed as a complete intersection of two quadric surfaces which are tangent at $p_1$, or as a projection of a rational normal quartic curve $C$ in $\mathbb{P}^{4}$ from a point lying on a secant line or an a tangent line of $C$.