1. **Problem statement:** Find the point(s) on the curve $x^2 - y^2 = 4$ closest to the point $(6,0)$. 2. **Formula and approach:** The distance squared from a point $(x,y)$ on the curve to $(6,0)$ is given by $$D^2 = (x-6)^2 + (y-0)^2 = (x-6)^2 + y^2.$$ Minimizing $D^2$ is equivalent to minimizing the distance. 3. **Use the constraint:** From the curve equation, $$x^2 - y^2 = 4 \\ \Rightarrow y^2 = x^2 - 4.$$ Substitute into $D^2$: $$D^2 = (x-6)^2 + (x^2 - 4) = (x-6)^2 + x^2 - 4.$$ 4. **Simplify:** $$D^2 = (x^2 - 12x + 36) + x^2 - 4 = 2x^2 - 12x + 32.$$ 5. **Minimize $D^2$ with respect to $x$:** Take derivative: $$\frac{dD^2}{dx} = 4x - 12.$$ Set to zero: $$4x - 12 = 0 \\ \Rightarrow x = 3.$$ 6. **Find corresponding $y$ values:** From the curve: $$y^2 = 3^2 - 4 = 9 - 4 = 5 \\ \Rightarrow y = \pm \sqrt{5}.$$ 7. **Answer:** The points on the curve closest to $(6,0)$ are $$\boxed{(3, \pm \sqrt{5})}.$$ 8. **Check options:** This corresponds to option (C).