If `alpha, beta` are the roots of the equation `x^(2)-7x+1=0` then value of
`{(1)/((alpha-7)^(2))+(1)/(beta-7)^(2))} - {((alpha^(2)+1)/(beta^(2)+1))+((beta^(2)+1)/(alpha^(2)+1))}` is

Options

A. `0`
B. `1`
C. `47`
D. `17`

Answer

A

Explanation

If `alpha , beta` are the ………..
`alpha^(2)-7alpha+1=0 implies alpha(alpha-7)= -1`
`implies alpha - 7 = -(1)/(alpha)` …..(i)
Similary `beta - 7 = -(1)/(beta)` …..(ii)
`:. (1)/((alpha-7)^(2))+(1)/((beta-7)^(2)) = (-alpha)^(2)+(-beta)^(2)`
`= alpha^(2)+beta^(2)` .....(iii)
Also , `alpha^(2)-7alpha+1=0`
`implies alpha^(2)+1=7alpha`
Similarly `beta^(2)+1=7beta`
`:. ((alpha^(2)+1)/(beta^(2)+1)) + ((beta^(2)+1)/(alpha^(2)+1))`
`= ((7alpha)/(7beta))+((7beta)/(7alpha))`
`= (alpha)/(beta)+(beta)/(alpha) = (alpha^(2)+beta^(2))/(alpha beta)`
`= alpha^(2)+beta^(2)` (be cause `alpha beta` = Product of roots `= 1` . ....(iv)
`:. (alpha^(2)+beta^(2))-(alpha^(2)+beta^(2))=0`