Parallel and perpendicular, part 2

Calculus Level 4

\[\begin{cases} x^2+y^2=1 \\ x^2-2xy+3y^2 =k\end{cases} \]

The above two curves touch each other at \(k=a\) and \(k=b\) where \(a>b\).

Let the area enclosed between the two curve be \(M\) and \(m\) for the case \(k=a\) and \(k=b\) respecively. Find the value of \(\left\lfloor{\dfrac{1000M}{m}}\right\rfloor\).

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