2005 AMC 12B Problems/Problem 8

Problem

For how many values of $a$ is it true that the line $y = x + a$ passes through the vertex of the parabola $y = x^2 + a^2$ ?

$\mathrm{(A)}\ 0      \qquad \mathrm{(B)}\ 1      \qquad \mathrm{(C)}\ 2      \qquad \mathrm{(D)}\ 10      \qquad \mathrm{(E)}\ \text{infinitely many}$

Solution

We see that the vertex of the quadratic function $y = x^2 + a^2$ is $(0,\,a^2)$. The y-intercept of the line $y = x + a$ is $(0,\,a)$. We want to find the values (if any) such that $a=a^2$. Solving for $a$, the only values that satisfy this are $0$ and $1$, so the answer is $\boxed{\mathrm{(C)}\ 2}$

See also

2005 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS