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) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }1\qquad\textbf{(D) }2\qquad\textbf{(E) }4