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If the sum of the roots of the quadratic equation `a x^2+b x+c=0` is equl to the sum of the squares of their reciprocals, then prove that `a/c , b/a a n d c/b` are in H.P.

Text Solution

Verified by Experts

Let `alpha,beta` be roots of `ax^(2)+bx+c=0`. Then
`alpha+beta=1/(alpha^(2))+1/(beta^(2))=(beta^(2)+alpha^(2))/(alpha^(2)beta^(2))`
`rArralpha+beta=((alpha+beta)^(2)-2alphabeta)/(alpha^(2)beta^(2))`
`rArr(-b)/a=((-b//a)^(2)-2c/a)/((c//a)^(2))`
or `(-b)/a=(b^(2)-2ac)/c^(2)`
or `-bc^(2)=ab^(2)-2a^(2)c`
or `2a^(2)c=ab^(2)+bc^(2)`
or `(2a)/b=b/c+c/a`
`rArrc/a,a/b,b/c` are in A.P.
`rArra,c,b/a,c/b` are in H.P.
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