If p and q are the roots of the equation #ax^2+bx+c=0#,find the value of #1/(ap^2+c)^2+1/(aq^2+c)^2#?

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#1/(ap^2+c)^2+1/(aq^2+c)^2?#

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Answer 1 · 2018-01-05T08:56:24

Given that p and q are the roots of the equation #ax^2+bx+c=0#
So
Sum of roots
#p+q=-b/a#

Product of roots

#pq=c/a#

#ap^2+bp+c=0#

#=>ap^2+c=-bp#

and
#aq^2+bq+c=0#

#=>aq^2+c=-bq#

the value of #1/(ap^2+c)^2+1/(aq^2+c)^2#

#=1/(-bp)^2+1/(-bq)^2#

#=1/b^2(1/p^2+1/q^2)#

#=1/b^2(p^2+q^2)/(pq)^2#

#=1/b^2((p+q)^2-2pq)/(pq)^2#

#=1/b^2((-b/a)^2-2(c/a))/(c/a)^2#

#=(b^2-2ca)/(bc)^2#