Question

If the sum of roots is equal to the squares of their reciprocals of a quadratic equation:- ax square+bx+c=0, then prove that bc square, ac square, ab square are in A.P

Answer 1

bc² , ca²  & ab² are in AP If the sum of roots = sum of squares of their reciprocals

Step-by-step explanation:

Correction :  bc², ca², ab² are in A.P

ax² + bx + c = 0

let say α , β  are the roots

then

α + β   =  1/α²  + 1/β²

LHS =

α + β  

= - b/a

RHS

= 1/α²  + 1/β²  

= (β²   + α²)/α²β²

= ((α + β)² - 2αβ)/α²β²

= ((α + β)² - 2αβ)/(αβ)²

αβ = c/a

= ( (-b/a)² - 2(c/a) ) /(c/a)²

=  (b² - 2ac)/c²

- b/a  = (b² - 2ac)/c²

=> -bc²  = ab²  - 2a²c

=> 2a²c = ab² + bc²

hence  bc² , a²c  & ab² are in AP

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