Problem:
Which of the following describes the set of values of for which the curves and in the real -plane intersect at exactly points?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Solving the second equation for gives , and substituting into the first equation gives . The polynomial in can be factored as , so the solutions are and . (Alternatively, the solutions can be obtained using the quadratic formula.) The corresponding equations for are and . The second equation always has the solution , corresponding to the point of tangency at the vertex of the parabola . The first equation has solutions if and only if , corresponding to the symmetric intersection points of the parabola with the circle. Thus the two curves intersect at points if and only if .
Substituting the value for from the second equation into the first equation yields
which is equivalent to
The first factor gives the solution , and the second factor gives other solutions if and no other solutions if . Thus there are solutions if and only if .
The problems on this page are the property of the MAA's American Mathematics Competitions