11. \( x=1-y^{2}, x=y^{2}-1 \) 11. The curves intersect when \( 1-y^{2}=y^{2}-1 \Leftrightarrow 2=2 y^{2} \Leftrightarrow y^{2}=1 \Leftrightarrow y= \pm 1 \).

When substituting \( y = 1 \) and \( y = -1 \) back into either equation to find \( x \), we find that both yield \( x = 0 \). So, the points of intersection are \( (0, 1) \) and \( (0, -1) \). These points represent locations where the two curves meet in a beautifully symmetrical way, showcasing not only the elegance of algebra but also the unity of mathematical concepts intersecting! Curves like these often appear in various fields, including physics and engineering, where they can represent trajectories or shapes of physical bodies. Understanding where they intersect is crucial for applications like optimization, where determining the best conditions or parameters often hinges on such analyses. So, next time you graph equations, remember you're not just working with numbers, but unraveling the dance of curves in the universe!