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2005 AMC 12B Problem 8

Problem 8 of 25EasierAlgebraGeometry

For how many values of aa is it true that the line y=x+ay = x + a passes through the vertex of the parabola y=x2+a2?y = x^2 + a^2?

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Solution

The vertex of the parabola y=x2+a2y = x^2 + a^2 is (0,a2).(0, a^2). The line y=x+ay = x + a passes through it exactly when a2=0+a,a^2 = 0 + a, that is a2−a=0.a^2 - a = 0. This gives a=0a = 0 or a=1,a = 1, so there are 22 values. Thus, the correct answer is C.

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Concepts: parabola · quadratic

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.