A Planar Cut Through an Ellipsoid

Geometry Level 5

An ellipsoid with center \((0,0,0)\) and semi-axes of lengths 10, 15, and 30 along the \(x, y,\) and \(z\) directions, respectively, is cut by a plane whose equation is \( x + 3y + 2z = 40 \).

The intersection of the ellipsoid and the plane is an ellipse.

Find its area.

This problem is original.

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