Simple bounds with best possible accuracy for ratios of modified Bessel functions

The best bounds of the form $B(\alpha,\beta,\gamma,x)=(\alpha+\sqrt{\beta^2+\gamma^2 x^2})/x$ for ratios modified Bessel functions are characterized: if $\alpha$, $\beta$ and $\gamma$ chosen in such a way that $B(\alpha,\beta,\gamma,x)$ is sharp approximation $\Phi_{\nu}(x)=I_{\nu-1} (x)/I_{\nu}(x)$ as $x\rightarrow 0^+$ (respectively +\infty$) graphs $\Phi_{\nu}(x)$ tangent at some $x=x_*>0$, then an upper lower) bound any positive $x$, it possible $x_*$. same true ratio $\Phi_{\nu}(x)=K_{\nu+1} (x)/K_{\nu}(x)$ but interchanging lower (and with slightly more restricted range $\nu$). Bounds maximal accuracy $0^+$ $+\infty$ recovered limits $x_*\rightarrow +\infty$, these cases coefficients have simple expressions. For case finite $x_*$ we provide uniparametric families which close to optimal retain their confluence properties.