Original Problem: For a given $y$ and $n$, I want to find all $x_i$'s that satisfy
\begin{equation} \frac{x_1^2}{\sum_{i=1}^{n}{x_i^2}}=y \tag{1} \end{equation}
while satisfying following constraints
- $x_i\geq0 \forall i=1,\cdots,n$
- $y \in [1/n,1]$
From geometry, when
1) $n=2$, solution to (1) comes from equation of circle of radius=1 i-e $x_1=\cos\theta$ and $x_2=\sin\theta$
2) $n=3$, solution to (1) comes from equation of sphere of radius=1 i-e $x_1^2+x_2^2+x_3^2=1$
But how do I find for $n\geq4$? I thought of do following
Reduced Problem: let $\sum_{i=1}^{n}{x_i^2}=1$. This implies that $y=+\sqrt{x_1}$. Moreover,
\begin{equation} x_2^2+x_3^2+\cdots+x_n^2 = 1-y \tag{2} \end{equation}
But how should I proceed from here?
The original equation can be written
$$\left(\frac{1-y}y\right)x_1^2=x_2^2+\cdots +x_n^2.$$
This is the equation of a cone (more precisely, its restriction to the positive quadrant).