From a point ‘P’ three normals are drawn to the parabola $y^{2} = 4 x$ such that two of them make angles with the abscissa axis, the product of whose tangents is 2. Suppose the locus of the point ‘P’ is a part of a conic C. Now a circle S = 0 is described on the chord of the conic ‘C’ as diameter passing through the point (1, 0) and with gradient unity. Suppose (a, b) are the coordinates of the centre of this circle. If $L_{1} a n d L_{2}$ are the two asymptotes of the hyperbola with length of its transverse axis 2a and conjugate axis 2b(principal axes of the hyperbola along the coordinate axes) then answer the following questions :

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