The \(x\) - and \(y\)-axes are the asymptotes of a hyperbola that passes through the point \((2,2)\). Its equation is a) \(x^{2}-y^{2}=0\) b) \(x y=4\) c) \(y^{2}-x^{2}=0\) d) \(x^{2}+y^{2}=4\)

The general equation of a hyperbola with the center at the origin and the asymptotes coinciding with the x and y-axes is given by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). In this case, since the asymptotes are the x- and y-axes, we have a standard hyperbola with a = b.

We'll now test all given equations to see if they represent a hyperbola with the specified asymptotes and pass through the point (2,2). a) \(x^2 - y^2 = 0\) The equation is symmetrical about the line y = x, but it does not represent a hyperbola. Moreover, it does not pass through the point (2,2). b) \(xy = 4\) This equation represents a hyperbola, but its asymptotes do not coincide with the x-axis and y-axis. However, it does pass through the point (2,2). c) \(y^2 - x^2 = 0\) The equation is symmetrical about the line y = -x, but it does not represent a hyperbola. Moreover, it does not pass through the point (2,2). d) \(x^2 + y^2 = 4\) This equation represents a circle with radius 2, rather than a hyperbola. Even though it passes through the point (2,2), it is not the correct choice.

Out of all given equations, only option b) \(xy = 4\), represents a hyperbola that passes through the point (2,2). Although option b) does not have asymptotes coinciding with the x-axis and y-axis, it is the best choice among the given options. Therefore, the correct answer is: b) \(xy = 4\)