247. Transform the equation of the hyperbola \( y = \frac{2x - 3}{x - 1} \) by a parallel translation of the axes, if the new origin is taken to be the point \( O_{1}(1; 2) \). Find the vertices of the hyperbola, the equations of the real and imaginary axes of symmetry in the new coordinate system, and in the old one.
Solution. We will use formulas (4). Substituting $x=X+1, y=Y+2$ into the given equation, we get:

$$
Y+2=\frac{2 X+2-3}{X+1-1}
$$

from which

$$
Y+2=2+\frac{-1}{X} \text { or } Y=-\frac{1}{X}
$$

In the new coordinate system $X O_{1} Y$, the branches of the hyperbola are located in the II and IV quadrants $(m=-1<0)$. The coordinates of the vertices of the equilateral hyperbola satisfy the condition $|X|=|Y|=\sqrt{|m|}=\sqrt{|-1|}=1$. Therefore, taking into account the quadrants, the vertices of the sought hyperbola are $A(-1 ; 1)$ and $A_{1}(1 ;-1)$, and in the original coordinate system, $A(0 ; 3)$ and $A_{1}(2 ; 1)$.

The equation of the real axis of symmetry is the bisector of the II and IV coordinate angles

$$
Y=-X
$$

in the original coordinate system

$$
y-2=-(x-1) \text { or } x+y-3=0
$$

The equation of the imaginary axis in the new coordinate system

$$
Y=X
$$

in the original coordinate system

$$
y-2=x-1 \text { or } x-y+1=0
$$