How to solve quadratic equations with three variable?

Question

_{Previously posted at MathOverflow}

---

Suppose we have $p$ quadratic equations \begin{eqnarray} \left\{ \begin{array}{lr} \mathbf{x}^\top \mathbf{C}_1 \mathbf{x} = 1, \\ \mathbf{x}^\top \mathbf{C}_2 \mathbf{x} = 1, \\ \quad\quad \vdots\\ \mathbf{x}^\top \mathbf{C}_p \mathbf{x} = 1, \\ \end{array} \right. \label{eq:quadEqus} \end{eqnarray} where $\mathbf{x} \in \mathbb{R}^3$, $\mathbf{C}_1 \sim \mathbf{C}_p \in\mathbb{R}^{3 \times 3}$ are positive definite symmetric matrices.

Suppose $\mathbf{x} = [a,b,c]^\top$, and $\mathbf{y} = [a^2,b^2,c^2, ab, ac, bc]^\top$, it is known that the above quadratic equations can be written as \begin{equation} \mathbf{Ay} = \mathbf{1}, \label{eq:linearsolver} \end{equation} where $\mathbf{A} \in \mathbb{R}^{p \times 6}$. If $p \geq 6$, we can obtain $\mathbf{y}$ and further estimate $\mathbf{x}$.

Questions:

1. Is there any methods besides $\mathbf{Ay}=\mathbf{1}$ to solve the above quadratic equations?
2. What is the minimal value of $p$ to guarantee a unique solution of $\mathbf{x}$ of the above quadratic system?

Thanks.