Algebra, Hyperbola.

Your post above looks as though it is a somewhat scrambled version of my previous post. But as I read the original question, it wants the mathematical results translated into descriptions in terms of the hypothesized roads and needs to distinguish between the double intersections that do and do not join the two curved roads.

This is actually a very clever exercise. A person who is good at mathematics can analyze problems, but such a person must usually also be good at explaining what the mathematical results mean for the mathematics to bear practical fruit.
 
"you did not explain what any of your mathematical conclusions meant in terms of the underlying engineering problem." What do you mean by this?
 
The actual problem as given was about roads. (Of course, the problem is super-stylized. That is typical of "word problems," which almost never reflect the complex details of problems in the real world.) Your answer was purely mathematical and said nothing about roads.

If you were a civil engineer given a problem about two roads that do not intersect, you might very well be interested in a new road that joined a specific location (the point) and the two pre-existing roads (the two branches of the hyperbola). The question was inviting you to say what your mathematical results meant. For most people, math is a tool to solve problems in the real world.
 
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