How to find certain quadratic curves over $\mathbb{Q}$

Question

Given a quartic curve C: $x^4+y^4=1$, how can I find a quadratic curve over $\mathbb{Q}$ intersecting $C$ at four points, while the intersection multiplicity of each point is 2?

Answer 1

The unit circle, $x^2 + y^2 = 1$ will intersect at four points with multiplicity 2. First of all, we know that they intersect at 4 points, $x = \pm 1, y = 0 $ and the same with x and y swapped.

We know from Bézout that there can be no more intersections (Fermat's Last Theorem also implies this, but that is a bit over the top for this problem), as long as we can show that each intersection is of order 2.

I shall leave that part of the problem to you.