# Difference between revisions of "2018 AMC 10A Problems/Problem 21"

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The other two intersection points have <math>x</math> coordinates <math>\sqrt{2a-1}</math>. We must have <math>2a-1> 0,</math> otherwise we are in the case where the parabola lies entirely above the circle (tangent to it at the point <math>(0,a)</math>). This only results in a single intersection point in the real coordinate plane. Thus, we see <math>a>\frac{1}{2}</math>. | The other two intersection points have <math>x</math> coordinates <math>\sqrt{2a-1}</math>. We must have <math>2a-1> 0,</math> otherwise we are in the case where the parabola lies entirely above the circle (tangent to it at the point <math>(0,a)</math>). This only results in a single intersection point in the real coordinate plane. Thus, we see <math>a>\frac{1}{2}</math>. | ||

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+ | (projecteulerlover) |

## Revision as of 17:25, 8 February 2018

Which of the following describes the set of values of for which the curves and in the real -plane intersect at exactly points?

## Solution

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 .

(projecteulerlover)