1985 AHSME Problems/Problem 19
Contents
Problem
Consider the graphs of and , where is a positive constant and and are real variables. In how many points do the two graphs intersect?
Solution 1
Substituting into the equation gives Now observe that since is positive, is also positive, so the square root will always give two distinct real values. Moreover, so , meaning that both solutions for are positive. Hence both solutions will give distinct values of (the positive and negative square roots), and each of these will correspond to a distinct point of intersection of the graphs, so there are points of intersection.
Solution 2
Firstly, note that is an upward-facing parabola (since ) whose vertex is at the origin. We now manipulate the equation of the second graph as follows: showing that it is a vertical (upward- and downward-opening) hyperbola with center and asymptotes and . It therefore remains to consider graphically where the parabola will intersect the hyperbola.
On the lower branch of the hyperbola, the maximum point is , which is above the vertex of the parabola. Therefore, by continuity and the symmetry of both the parabola and the hyperbola in the -axis, there are always exactly intersection points here.
For the top branch, as it approaches the asymptote , its slope also approaches that of this asymptote, which is . However, for any upward-opening parabola, the slope approaches infinity as does, so no matter how small is (i.e. how 'flat' the parabola is), the parabola will eventually overtake the hyperbola, giving a point of intersection with positive -coordinate. As above, symmetry gives another point of intersection with negative -coordinate, so that there are intersection points with this branch too.
Thus there are a total of intersection points.
See Also
1985 AHSME (Problems • Answer Key • Resources) | ||
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Followed by Problem 20 | |
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