Hyperbolic Conic Section

Geometry Level pending

A twin cone has the angle between its curved surface and its axis equal to \( \theta_c \). The axis of the twin cone system is along the z-axis, and its vertex is at the origin. A plane having the equation

\[ -\sin( \theta_p ) (x - x_0) + \cos(\theta_p) z = 0 \]

intersects with the twin cones. We are given that \( \theta_c = \cos^{-1} \left(\dfrac{4}{5}\right) \) and that \( \theta_p = \tan^{-1}( 2) \) and that \( x_0 = 5 \). With these values for \( \theta_c \) and \( \theta_p \), the intersection is a hyperbola. By attaching a reference frame \(x'\)-\(y'\) to the cutting plane and choosing its origin to be at the center of the hyperbola, and also choosing its orientation appropriately, with the \(x'\)-axis lying parallel to the \(xy\)-plane and the \(y'\)-axis running up the cutting plane, the resulting hyperbola has the following equation

\[ \dfrac{ y'^2 }{a^2} - \dfrac{ x'^2 }{b^2} = 1 \]

The center of the hyperbola is at the point \( (c, 0, d),\) for positive \( a,b,c,\) and \( d \). Find the value of \(a^2 + b^2 + c + d\).

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