Intersection of a surface and a line.

Question

I have a surface ($H$) that passes every corner points(one coordinate gets its maximum $1$ while others $0$), such as

$(1,0,...,0), (0,1,0,...,0),....,(0,...,1).$

H is characterized by

$A\sum_{i=1}^{n}(\beta_i)^2+\sum_{i\neq j}^{n}(\beta_i)(\beta_j)=A$,

where $A\in[0,1)$.

The figure below illustrates the case for $n=2$.

I have a point $x$ on the boundary, the line(L) passes initial point $\mathbf{0}$ and $x$ intersects $H$ at point $d$. How to locates $d$(vector notation)?

Thanks in advance!