2022 AMC 12B 第 8 题

Question

方程

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

在坐标平面中的图像是什么？

What is the graph of

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

in the coordinate plane?

两条相交抛物线

two intersecting parabolas

两条不相交抛物线

two nonintersecting parabolas

两个相交圆

two intersecting circles

一个圆和一条双曲线

a circle and a hyperbola

一个圆和两条抛物线

a circle and two parabolas

移项得到

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

。

Move everything to one side to get

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

作为平方差分解，得到两个单独的方程。

Factor as a difference of squares, giving two separate equations

整理得

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

，即

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

。作平方差分解，得

(y^2 - 1 - x^2)(y^2 - 1 + x^2) = 0\text{。}

因此或者

y^2 - x^2 = 1

，这是一条双曲线；或者

x^2 + y^2 = 1

，这是一个圆。

所以正确答案是 D。

Rearranging,

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

so

(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.

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