1. The problem is to analyze the equation of the form $$x^2 - y^2 = 49$$. 2. This equation represents a hyperbola because it is in the form $$x^2/a^2 - y^2/b^2 = 1$$ where $$a^2 = 49$$ and $$b^2 = 49$$. 3. Rewrite the equation as $$\frac{x^2}{49} - \frac{y^2}{49} = 1$$. 4. From this, we identify $$a = 7$$ and $$b = 7$$. 5. The hyperbola opens along the x-axis because the positive term is $$x^2$$. 6. The vertices are at $$(\pm a, 0) = (\pm 7, 0)$$. 7. The foci are at $$(\pm c, 0)$$ where $$c = \sqrt{a^2 + b^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2}$$. 8. The asymptotes of the hyperbola are given by $$y = \pm \frac{b}{a} x = \pm x$$. 9. Summary: - Equation: $$\frac{x^2}{49} - \frac{y^2}{49} = 1$$ - Vertices: $$(\pm 7, 0)$$ - Foci: $$(\pm 7\sqrt{2}, 0)$$ - Asymptotes: $$y = \pm x$$