Moving Ellipse

If the curve $2y^2 + x^2 = 64$ is moved $5$ units to the right, its new equation is $2y^2 + (x - 5)^2 = 64$ . The point of intersection $(x, y)$ is the solution to these two equations, which by subtraction gives $x^2 - (x - 5)^2 = 0$ which solves to $x = \frac{5}{2}$ , and by substitution gives $2y^2 + (\frac{5}{2})^2 = 64$ which solves to a positive value of $y = \frac{\sqrt{462}}{4}$ .

Therefore, $\lfloor x \times y \rfloor = \lfloor \frac{5}{2} \times \frac{\sqrt{462}}{4} \rfloor = \boxed{13}$ .