16 Let \( (H) \) be the hyperbola representing the function defined by \( f(x)=\frac{m x-3}{3 x-m} \), where \( m \) is is a real parameter other than \( 3 .(D) \) is the line of equation \( y=-x+4 \). \( 1^{\circ} \) Find the coordinates of the center \( I \) of \( (H) \) and deduce the set of points \( I \) when \( m \) varies.

The hyperbola \( (H) \) is given by the function \( f(x) = \frac{mx - 3}{3x - m} \). To find the center of the hyperbola, we need to express it in the form of the standard hyperbola equation. The center of a hyperbola generally occurs at the average of its asymptotes. Firstly, we can determine the vertical and horizontal asymptotes of the hyperbola by finding the values of \( x \) and \( y \) where the function is undefined. The vertical asymptote occurs when the denominator is zero, so we set \( 3x - m = 0 \), giving us \( x = \frac{m}{3} \). For the horizontal asymptote, we investigate the behavior of \( f(x) \) as \( x \) approaches infinity. We can see that as \( x \rightarrow \infty \), the dominant terms in \( f(x) \) (which are \( mx \) in the numerator and \( 3x \) in the denominator) lead us to the horizontal asymptote \( y = \frac{m}{3} \). Thus, the center \( I \) of the hyperbola is located at the point \( \left( \frac{m}{3}, \frac{m}{3} \right) \). As \( m \) varies, this point traces the line \( y = x \) in the coordinate plane. Therefore, the set of points \( I \) when \( m \) varies is the line defined by \( y = x \), excluding the point where \( m = 3 \), which results in an indeterminate form. In conclusion, the coordinates of the center \( I \) are \( \left( \frac{m}{3}, \frac{m}{3} \right) \), and as \( m \) changes (except when \( m = 3 \)), \( I \) moves along the line \( y = x \).