3D and Limits

Geometry Level 4

The incenter \(I\) of the triangle \(PQR\) is the foot of the normal from the point \(M = (1,2,6) \) to the \(xy\)-plane, where \(P,Q,R\) are the feet of altitudes of an isosceles triangle \(ABC\) whose vertex is \(A\) and base \(BC\) of 6 unit length.

Let \( \displaystyle \lim_{A\to {\frac \pi 2}^+} \dfrac{e^v - e^k}{\sqrt{1- \sin A}} = \dfrac{e^k}L \) for integer \(k\), where \(v\) is the volume of the tetrahedron \(MIBC\).

Find the value of \( \dfrac1{k^2 L^2 } \).

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