Question

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?

$a = \dfrac14$

$\dfrac14 \lt a \lt \dfrac12$

$a \gt \dfrac14$

$a = \dfrac12$

$a \gt \dfrac12$

Answer: E

Concepts:parabola circle substitution

Difficulty rating: 1820

Solution:

Substitute $y=x^2-a$ into $x^2+y^2=a^2$ . This gives $x^2+(x^2-a)^2=a^2$ , so $x^2(x^2-(2a-1))=0$ .

The factor $x^2=0$ always gives the single point $(0,-a)$ . The other factor gives two additional real points exactly when $2a-1>0$ .

There are exactly three intersection points when $a>\dfrac12$ . Thus, E is the correct answer.

2018 AMC 10A Problem 21

Answer: E

Difficulty rating: 1820

Solution:

Problem 21 in Other Years