for a non-degenerate integral quadratic form $f(x_1, \dots , x_d)$ in 5 (or more) variables, we prove an optimal strong approximation theorem. fix any compact subspace $\omega\subset\mathbb{r}^d$ of the affine quadric $f(x_1,\dots,x_d)=1$. suppose that we are given a small ball $b$ of radius $00$. finally assume that we are given an integral vector $(\lambda_1, \dots, \lambda_d) $ mod $m$. then we show that there exists an integral solution $x=(x_1,\dots,x_d)$ of $f(x)=n$ such that $x_i\equiv \lambda_i \text{ mod } m$ and $\frac{x}{\sqrt{n}}\in b$, provided that all the local conditions are satisfied. we also show that 4 is the best possible exponent. moreover, for a non-degenerate integral quadratic form $f(x_1, \dots , x_4)$ in 4 variables we prove the same result if $n\geq (r^{-1}m)^{6+\epsilon}$ and some non-singular local conditions for $n$ are satisfied. based on some numerical experiments on the diameter of lps ramanujan graphs, we conjecture that the optimal strong approximation theorem holds for any quadratic form $f(x)$ in 4 variables with the optimal exponent $4$.