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.