I found this question: Chernoff style concentration bound for ratio of variables. I want to ask if we get similar thing for the ratio of the sum and the one Gaussian variable.

Given i.i.d. Gaussian random variables $X_1,\dots, X_k$ with $N(0, 1)$. Fix $\epsilon\in (0,1)$, can prove that for any $\delta>0$ there exists $1\le k=k(\epsilon, \delta)$ (some number) so that $$ P\left(\frac{X_1^2+X_2^2+\dots+X_k^2}{X_1^2}>\frac{1}{\epsilon^2}\right)>1- \delta ? $$

Can we find such $k$?

In these 2020 slides by Andrew Nobel, for $Y\sim \chi_k^2$ where $(Y=\sum_{I=1}^k X_i^2)$, for $t\in (0,1)$ $$ P(Y\ge (1+\epsilon)k)\le \exp(-k(t^2-t^3)/4). $$