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

Answer

1

Explanation

Let `2010 = n`
`:. n^(2)(1-pq)-n(p^(2)-q^(2))-(1+pq) = 0`
`(n^(2)-1)=pq(n^(2)+1)+n(p^(2)-q^(2))….(1)`
`d=(p-q)/(pq(n+1)):.(1)/(h_(1))=(1)/(p)+d,(1)/(h_(2))=(1)/(q)-d`
`h_(1)-h_(n)=(1)/((1)/(p)+d)-(1)/((1)/(q)-d)=(P)/(1+pd)-(q)/(1-qd)`
`(p)/(1+(p(p-q))/(pq(n+1)))-(q)/(1+(q(p-q))/(pq(n+1)))`
`= (pq(p-q)(n^(2)-1))/((qn+p)(pn+q))`
`=(pq(p-q)[pq(n^(2)+1)+n(p^(2)+q^(2))])/((qn+p)(pn+q))`
`:. (h_(1) - h_(n))/(pq(p-q)) = 1`