248. The equation of the hyperbola $y=\frac{1-3 x}{2 x-1}$ can be transformed using a parallel translation of the coordinate axes to the form $X Y=m$. Plot this hyperbola.
Solution. In this equation, by dividing the numerator by the denominator, we get:

$$
\begin{gathered}
y=-\frac{3}{2}-\frac{\frac{1}{4}}{x-\frac{1}{2}} \text { or } y+1.5=\frac{-0.25}{x-0.5}, \\
\text { or } Y=-\frac{0.25}{X}, \text { or } X Y=-0.25,
\end{gathered}
$$

where $X=x-0.5$ and $Y=y+1.5$.

The origin of the new coordinate system is at the point $O_{1}(0.5, -1.5)$. Since $m=-0.25<0$, the branches of the hyperbola are located in the II and IV quadrants. Let's find the points of intersection of the hyperbola with the old coordinate axes. If in this equation $x=0$, then $y=-1$; if $y=0$, then $x=\frac{1}{3}$, hence the points of intersection are: $B(0, -1)$ and $C\left(\frac{1}{3}, 0\right)$. Now let's plot the hyperbola (Fig. 31).