2018 AMC 12A Problem 16

Problem 16 of 25IntermediateAlgebra Geometry

Which of the following describes the set of values of

a

for which the curves

x^2 + y^2 = a^2

and

y = x^2 - a

in the real

xy

-plane intersect at exactly

3

points?

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Solution

Substituting

x^2 = y + a

into

x^2 + y^2 = a^2

gives

y^2 + y + (a - a^2) = 0,

which factors as

(y + 1 - a)(y + a) = 0,

y = a - 1

y = -a.

These correspond to

x^2 = 2a - 1

and

x^2 = 0.

The equation

x^2 = 0

always gives the single point

(0, -a),

the vertex of the parabola. The equation

x^2 = 2a - 1

gives two more points exactly when

2a - 1 \gt 0,

i.e.

a \gt \tfrac12.

So there are

3

intersection points precisely when

a \gt \tfrac12.

Thus, the correct answer is E.