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 this slide: https://nobel.web.unc.edu/wp-content/uploads/sites/13591/2020/10/Probability_Inequalities.pdf, for $Y\sim \chi_k^2$ (Y=\sum_{I=1}^k X_i^2), for $t\in (0,1)$ $$ P(Y\ge (1+\epsilon)k)\le \exp(-k(t^2-t^3)/4). $$