Cutting A Scaled Cone

Geometry Level 5

A cone with vertex at the origin and opening upward, with axis along the positive $$z$$-axis, is scaled (stretched) along the $$y$$-axis direction by a factor of 2. The angle between the surface of the unstretched cone and its axis, is $$20^{\circ}$$. A plane with equation $$\sqrt{2} x + \sqrt{2} y + 2 \sqrt{3} z = 100 \sqrt{3}$$ cuts though this stretched cone. The intersection between the plane and the stretched cone is an ellipse. Find the sum of the semi-minor and semi-major axes of this ellipse.