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 $\dfrac1{k^2 L^2 }$ .

The answer is 2.

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This is a solution by my friend

@Devansh Jain .