`pq ^(2) + q (p -1) -1=` ?

(p+q)^(2) = p^(2) + 2pq + q^(2) = 1 represents an equation used in

If 2010 is a root of x^(2)(1 - pq) - x(p^(2) + q^(2)) - (1 + pq) = 0 and 2010 harmonic mean are inserted between p and q then the value of (h_(1) - h_(2010))/(pq(p - q)) is

If 2010 is a root of x^(2)(1 - pq) - x(p^(2) + q^(2)) - (1 + pq) = 0 and 2010 harmonic mean are inserted between p and q then the value of (h_(1) - h_(2010))/(pq(p - q)) is

If (p + 2) (2q - 1) = 2pq - 10 and (p - 2) (2q - 1) = 2pq - 10, then what is pq equal to ?

(a) If r^(2) = pq , show that p : q is the duplicate ratio of (p + r) : (q + r) . (b) If (p - x) : (q - x) be the duplicate ratio of p : q then show that : (1)/(p) + (1)/(q) = (1)/(r) .