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

Ishankhare (talk | contribs) (Created page with "Which of the following describes the set of values of <math>a</math> for which the curves <math>x^2+y^2=a^2</math> and <math>y=x^2-a</math> in the real <math>xy</math>-plane i...") |
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\textbf{(E) }a>\frac12 \qquad | \textbf{(E) }a>\frac12 \qquad | ||

</math> | </math> | ||

+ | |||

+ | == Solution == | ||

+ | |||

+ | Substituting <math>y=x^2-a</math> into <math>x^2+y^2=a^2</math>, we get | ||

+ | <cmath> | ||

+ | x^2+(x^2-a)^2=a^2 \implies x^2+x^4-2ax^2=0 \implies x^2(x^2-(2a-1))=0 | ||

+ | </cmath> | ||

+ | Since this is a quartic, there are 4 total roots (counting multiplicity). We see that <math>x=0</math> always at least one intersection at <math>(0,-a)</math> (and is in fact a double root). | ||

+ | |||

+ | 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>. |

## 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 .