Conic Section

Geometry Level 5

A right circular cone with its vertex at the origin (0, 0, 0) and a vertical axis, is cut by the plane \( 3 x + 4 y + 5 z = 500 \). The angle of the vertex (between the axis of the cone and its curved surface) is \( 20^{\circ} \).

The intersection in this case is an ellipse. Find the sum of the lengths of its semi-minor and semi-major axes.

This problem is original.

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