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.

To analyze the hyperbola \( (H) \) defined by the function \( f(x) = \frac{mx - 3}{3x - m} \), where \( m \) is a real parameter (with \( m \neq 3 \)), we can determine the coordinates of its center \( I \) and describe the locus of \( I \) as \( m \) varies. ### 1. Finding the Coordinates of the Center \( I \) A rational function of the form \( y = \frac{ax + b}{cx + d} \) typically represents a hyperbola with: - **Vertical Asymptote:** \( x = -\frac{d}{c} \) - **Horizontal Asymptote:** \( y = \frac{a}{c} \) The **center** of the hyperbola is the intersection point of these two asymptotes. For the given function \( f(x) = \frac{mx - 3}{3x - m} \), we can identify the coefficients as: - \( a = m \) - \( b = -3 \) - \( c = 3 \) - \( d = -m \) **Calculating the Asymptotes:** - **Vertical Asymptote:** \( x = -\frac{d}{c} = -\frac{-m}{3} = \frac{m}{3} \) - **Horizontal Asymptote:** \( y = \frac{a}{c} = \frac{m}{3} \) Therefore, the center \( I \) of the hyperbola \( (H) \) is located at: \[ I = \left( \frac{m}{3}, \frac{m}{3} \right) \] ### 2. Describing the Set of Points \( I \) as \( m \) Varies As the parameter \( m \) varies over all real numbers except \( m \neq 3 \), the coordinates of the center \( I \) trace out points of the form \( \left( \frac{m}{3}, \frac{m}{3} \right) \). This set of points lies on the line defined by: \[ y = x \] since both the \( x \) and \( y \) coordinates of \( I \) are equal and scale proportionally with \( m \). Thus, **the locus of all centers \( I \)** as \( m \) varies is the straight line \( y = x \). ### **Final Answer** 1°) For each real m ≠ 3, the center I is (m⁄3, m⁄3).   2°) As m varies, I moves along the line y = x. Thus, all centers lie on the straight line y = x.