2022 AMC 12B Problem 8

Attempt Problem 8 of the 2022 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 2022 AMC 12B solutions, or check the answer key.

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

What is the graph of

y^4 + 1 = x^4 + 2y^2

in the coordinate plane?

two intersecting parabolas

two nonintersecting parabolas

two intersecting circles

a circle and a hyperbola

a circle and two parabolas

Move everything to one side to get

(y^2 - 1)^2 = (x^2)^2

Factor as a difference of squares, giving two separate equations

Rearranging,

y^4 - 2y^2 + 1 = x^4,

(y^2 - 1)^2 = (x^2)^2.

This factors as

(y^2 - 1 - x^2)(y^2 - 1 + x^2) = 0.

Thus either

y^2 - x^2 = 1,

which is a hyperbola, or

x^2 + y^2 = 1,

which is a circle.

Thus, the correct answer is D.

Problem 8 in other years