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If the quadratic equation a^(2)(b^(2) -c...

If the quadratic equation `a^(2)(b^(2) -c^(2)) x^(2) +b^(2) (c^(2) -a^(2)) x+c^(2) (a^(2) -b^(2)) =0` has real and equal roots, than ` a^(2) , b^(2) ,c^(2) `are
(1) A.P. (2) (G.P. (3) H.P. (4) A.G.P.

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we observe that the sum of coefficients of the given equation.
`a^(2)(b^(2)-c^(2))x^(2) +b^(2) (c^(2) -a^(2))x+c^(2)(a^(2) -b^(2))=0` is zero
i.e.`a^(2) (b^(2) -c^(2))+b^(2)(C^(2)-a^(2))+c^(2) (a^(2)-b^(2))=0`
Therefore roots are 1 and `(c^(2)(a^(2)-b^(2)))/(a^(2)(b^(2) -c^(2)))`
in case of equal roots .
`1= (c^(2)(a^(2)-b^(2)))/(a^(2)(b^(2) -c^(2)))`
` Rightarrow a^(2)b^(2) -a^(2)c^(2)=a^(2)c^(2)-b^(2)c^(2)`
` Rightarrow a^(2)b^(2) +b^(2)c^(2) =2a^(2)c^(2)`
` Rightarrow 1/c^(2) + 1/a^(2) =2/b^(2)`
` Rightarrow a^(2) , b^(2),c^(2)` are in H.P.
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