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
.