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Answers
Given---> Sum of roots of given quadratic eqution is equal to the sum of squares of their reciprocal and equation is ax² + bx + c = 0
Solution---> We know that if x , y , z are in AP then , x + z = 2y , in other words it is the condition that x, y , z are in AP.
ATQ , quadratic equation is
ax² + bx + c = 0
It is a quadratic equation so it has two roots .
Let , roots of quadratic equation be α and β .
Now we know that,
Sum of roots = - coefficient of x / coefficient of x²
=> α + β = - b / a
Product of roots = constant term / coefficient of x²
=> αβ = c / a
Now , ATQ,
Sum of roots = Sum of squares of reciprocal of
roots
=> α + β = ( 1 / α² + 1 / β² )
=> α + β = ( α² + β² ) / α² β²
=> α² β² ( α + β ) = α² + β²
=> ( αβ )² ( α + β ) = α² + β² + 2αβ - 2αβ
=> ( αβ )² ( α + β ) = ( α + β )² - 2αβ
Putting α + β = - b / a and α β = c / a
=> ( c / a )² ( - b / a )= ( - b / a )² - 2 ( c / a )
=> - ( c² / a² ) ( b/ a) = ( b² / a² ) - 2 ( c / a )
=> - b c² / a³ = ( b² - 2ac ) / a²
=> - a²bc² = a³ ( b² - 2ac )
=> -a² b c² = a² a¹ ( b² - 2ac )
a² cancel out from both sides and we get
=> - b c² = a ( b² - 2ac )
=> - bc² = ab² - 2a²c
=> 2 a² c = ab² + bc²
=> b c²+ a b² = 2 c a²
It is the required condition So ab² , ca² and ab² are in AP