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.