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An equatin of the family of circles passing through a given pair of points (x_(1), y _(1))and (x _(2), y _(2)) can be taken ad (x - x _(1)) (x -x _(2))+ (y - y_(1)) (y-y_(2))+ k |{:(x,y,1),(x_(1), y _(1), 1),(x_(2), y_(2), 1):}|=0, k being a real parameter. If a member of this family satisfies some other condition then that enables us to determine k and hence the number. The number of values of lamda in R for which there exists exactly one circle passing through the point (2, -3) and (lamda, 2 lamda -1) and touching the line 16x-2y + 27=0, is

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infinitely MANY

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