12.48 The variable $y$ is inversely proportional to the variable $x$. Find the coefficient $k$ of inverse proportionality and fill in the table:

| $x$ |  | 0.1 | 9.6 |
| :---: | :---: | :---: | :---: |
| $y$ | 30 |  | 3.05 |

On the graph of the given inverse proportionality, find the point closest to the origin $O(0 ; 0)$.
12.48 According to the condition, $y=\frac{k}{x}$ or $k=x y$; thus, $k=9.6 \cdot 3.05 = 29.28$. If $y=30$, then $x=\frac{29.28}{30} \approx 0.98$; if $x=0.1$, then $y=\frac{29.28}{0.1}=292.8$. The distance from the origin $(0 ; 0)$ to the point $(x ; y)$ lying on the hyperbola is $\sqrt{x^{2}+y^{2}}$. Therefore, we need to investigate the function $f(x)=\sqrt{x^{2}+\frac{k^{2}}{x^{2}}}$ for extremum, where $x \in(0, \infty)$. We find $f^{\prime}(x)=\frac{2 x-\frac{k^{2}}{x^{4}} \cdot 2 x}{2 \sqrt{x^{2}+\frac{k^{2}}{x^{2}}}}=\frac{x\left(1-\frac{k^{2}}{x^{4}}\right)}{\sqrt{x^{2}+\frac{k^{2}}{x^{2}}}}$.

Thus, $f^{\prime}(x)=0$ if $1-\frac{k^{2}}{x^{4}}=0$, or $x^{4}=k^{2}$, from which $x=\sqrt{|k|}$. For $k>0$, we have $y=\frac{k}{\sqrt{k}}=\sqrt{k}$, i.e., $x=y=\sqrt{k}$. Therefore, $x=y \approx 5.4$.

Answer: $k=29.28 ; (5.4 ; 5.4)$.