# 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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