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Consider the hyperbola (x^(2))/100 - (...

Consider the hyperbola
`(x^(2))/100 - (y^(2))/64 = 1 ` with foci at S and `S_(1)` ,where S lies on the positive x - axis ,Let P be a point on the hyperbola , in the first quadrant ,let `angleSPS_(1) = alpha ` , with `alpha lt pi/2 ` .The straight line passing through the point S and having the same slope as that of the tangent at P to the hyperbola , intersects the straight line `S_(1)P` at `P_(1)` .Let `delta ` be the distance of P from the straight line `SP_(1)` and `beta =S_(1)P and beta = S_(1)P` . Then the greatest integer less than or equal to `(betadelta)/9 sin alpha/2 ` is ________ .

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