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Q.
If $a, b, c, x$ are real numbers and equation $\left(a^2+b^2\right) x^2-2 b(a+c) x+\left(b^2+c^2\right)=0$ hảs equall roots. then $a, b, c$ are in-
Sequences and Series
Solution:
If $a, b, c, x \in R$ and $\left(a^2+b^2\right) x^2-2 b(a+c) x+\left(b^2+c^2\right)=0$
This is quadratic in $x$, for equal root $D =0$
$ \Rightarrow 4 b^2(a+c)^2=4\left(a^2+b^2\right)\left(b^2+c^2\right)$
$ \Rightarrow b^2 a^2+b^2 c^2+2 a c b^2=a^2 b^2+a^2 c^2+b^4+b^2 c^2$
$\Rightarrow b^4-2 a c b^2+a^2 c^2=0 $
$\Rightarrow\left(b^2-a c\right)^2=0 \Rightarrow b^2=a c $
$ \therefore a, b, c \text { are in G.P. }$