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2022 AMC 12B Problem 8

Problem 8 of 25EasierAlgebraGeometry

What is the graph of y4+1=x4+2y2y^4 + 1 = x^4 + 2y^2 in the coordinate plane?

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Solution

Rearranging, y4−2y2+1=x4,y^4 - 2y^2 + 1 = x^4, so (y2−1)2=(x2)2.(y^2 - 1)^2 = (x^2)^2. This factors as (y2−1−x2)(y2−1+x2)=0. (y^2 - 1 - x^2)(y^2 - 1 + x^2) = 0. Thus either y2−x2=1,y^2 - x^2 = 1, which is a hyperbola, or x2+y2=1,x^2 + y^2 = 1, which is a circle. Thus, the correct answer is D.

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Concepts: difference of squares · circle · hyperbola

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