2024 Mock AMC 12 - Problem 24

Let $\mathcal{P}$ be the image when the parabola $y=x^2$ is rotated $45^\circ$ clockwise around the origin. Then, the equation $x^2+y^2-\sqrt{2}x-\sqrt{2}y=k$ intersects $\mathcal{P}$ at more than $2$ points. The values of $k$ that satisfy this form a half-open interval with length $\ell$. What is $\ell$?

$\textbf{(A)}~\tfrac{1}{4}\qquad\textbf{(B)}~\tfrac{1}{2}\qquad\textbf{(C)}~1\qquad\textbf{(D)}~2\qquad\textbf{(E)}~4$