2005 AMC 12B Problem 8

Attempt Problem 8 of the 2005 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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?

0

1

2

10

infinitely many

The vertex of

y = x^2 + a^2

(0, a^2)

Substitute the vertex into the line to get

a^2 = a

The vertex of the parabola

y = x^2 + a^2

(0, a^2).

The line

y = x + a

passes through it exactly when

a^2 = 0 + a,

that is

a^2 - a = 0.

This gives

a = 0

a = 1,

so there are

2

values.

Thus, the correct answer is C.

Problem 8 in other years