2018 AMC 10A Problems/Problem 21
Which of the following describes the set of values of for which the curves and in the real -plane intersect at exactly points?
Substituting into , we get Since this is a quartic, there are 4 total roots (counting multiplicity). We see that always at least one intersection at (and is in fact a double root).
The other two intersection points have coordinates . We must have otherwise we are in the case where the parabola lies entirely above the circle (tangent to it at the point ). This only results in a single intersection point in the real coordinate plane. Thus, we see .