Where is the Vertex?

Geometry Level 5

A solid right circular cone is such that the angle between its axis and its surface is \( 20^{\circ} \). A plane cuts through the cone at a certain angle with its axis, thus generating an elliptical base. The cone is placed on its elliptical base on a flat surface, such that the center of the ellipse coicides with the origin of an XYZ reference frame with the XY plane lying along the flat surface, and the positive Z-axis pointing upward. The XY axes orientation is such that the major axis of the elliptical base is along the X-axis and the minor axis is along the Y-axis.If the equation of the edge of the base is given by

\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]

What are the coordinates of the vertex? Assume that the vertex is located at \((a, b, c)\) where \( a \ge 0 , b \ge 0, c \ge 0 \). Enter the value of \( a+b+c \) as your answer.

Hint: Use the formulas in the solution of this problem.


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