1.

The sum of the roots of the equation ax2+x+c=0 (where a and c are non-zero) is equal to the sun of the reciprocal of their squares. Then a, ca2, c2  are in (A) AP.  (B)GP (c) HP. (D)AGP

Answer»

SOLUTION: 
Let the roots of the equation be α and β.  Then  
       α +  β = 1/α²  +  1/β²

or,  α +  β = (α²+β²) / α²β²

or,  α +  β = ( α +  β)² - 2αβ   / α²β²

Putting the values of α +  β and αβ we get 

      -b/a =( (b²/a²) - 2c/a) / (c²/a²)

or,    -b/a = (b²- 2ac)/ c²

or,    -bc² = ab² - 2ca²

or    bc² + ab² = 2ca²

Hence bc² , ca², ab² are in A.P.


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