2022 AMC 12B Problem 8

What is the graph of

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

in the coordinate plane?

Show solution

Solution

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.

Tagged: difference of squares · circle · hyperbola