If \( p \) and \( q \) are the lengths of the perpendiculars from the origin on the lines, \( x cosec \alpha - y \,\text{sec}\, \alpha = k \cot 2\alpha \) and \( x \sin \alpha + y \cos \alpha = k \sin 2\alpha \) respectively, then \( k^{2} \) is equal to:
