1985 AHSME Problems/Problem 19
Consider the graphs and , where is a positive constant and and are real variables. In how many points do the two graphs intersect?
Solution 1: Algebra
Substitute into to get . Let , so that . By the quadratic formula, we have . Notice that since is positive, is also positive, so the square root is real. Also, , so , and both solutions to are positive. Thus, we have two possible values for , and taking both the positive and negative roots, we have possible values for , .
Solution 2: Graphing
Notice that is an upward-facing parabola with a vertex at the origin. We can manipulate into a recognizable graph:
This is just a vertical hyperbola! The center of the hyperbola is and it has asymptotes at and . Now we can see where a parabola would intersect this hyperbola. It would seem obvious that the parabola intersects the lower branch of the hyperbola twice. For a more rigorous argument, consider that since the y value is always positive, the entire parabola would have to be contained in that area between the lower branch and the x-axis, clearly a violation of the end behavior of approaching infinity.
As for the top branch, it may seem to make sense that a small enough would make a 'flat' enough parabola so that it doesn't intersect the top branch. However, this is not the case. Consider that as the branch approaches its asymptote, its slope also approaches that of its asymptote, which is . However, for any parabola, the slope approaches infinity as does, so eventually any upward-facing parabola will overtake the hyperbola on both sides, for intersection points on the top branch, too.
Thus, we have a total of intersection points, .
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