What conditions should vector x satisfy so that $\Vert [{x}_{2}+\alpha {x}_{1},\dots ,{x}_{n}+\alpha {x}_{1}]{\Vert}_{2}$ is bounded by a constant?

Suppose that $x=[{x}_{1},\dots ,{x}_{n}]$ is a vector with norm less than or equal to one $\Vert x{\Vert}_{2}^{2}\le 1$. Let $\alpha \in [0,1]$ and define the following vector

$y=[{x}_{2}+\alpha {x}_{1},\dots ,{x}_{n}+\alpha {x}_{1}]$

How can I find a non-trivial subset of x that, regardless of the value of α, would result in $\Vert y{\Vert}_{2}^{2}\le C$ where C is a constant that does not depend on n?

The above is satisfied when ${x}_{1}=\cdots ={x}_{n}$. For example, if $x=[1/\sqrt{n},\dots ,1/\sqrt{n}]$ then, we have $\Vert y{\Vert}_{2}^{2}=\frac{(1+\alpha {)}^{2}}{n}\times n=(1+\alpha {)}^{2}\le 4$. But I'm looking for a larger subset (or all x that satisfy the above conditions) than ${x}_{1}=\cdots ={x}_{n}$