Mathematics
Mathematics
Problem
Find the position of the point relative to the hyperbola .Intuitive explanation
The key is to treat the hyperbola equation like a test: substitute the point's coordinates and see whether the result is greater than, less than, or equal to the defining value. If the point satisfies the equation exactly, it's on the hyperbola; if the left side is larger, the point lies in a region "outside" the hyperbola's branches; if smaller, it's in the region between them. This tells you the point's geometric position without needing to sketch anything.
Intuitive explanation
The key is to treat the hyperbola equation like a test: substitute the point's coordinates and see whether the result is greater than, less than, or equal to the defining value. If the point satisfies the equation exactly, it's on the hyperbola; if the left side is larger, the point lies in a region "outside" the hyperbola's branches; if smaller, it's in the region between them. This tells you the point's geometric position without needing to sketch anything.