Question

If $a,b,c,x$ are real numbers and equation $\left(a^2+b^2\right)x^2-2b(a+c)x+\left(b^2+c^2\right)=0$ hảs equall roots. then $a,b,c$ are in-

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

Answer: (B)

If $a,b,c,x\in R$ and $\left(a^2+b^2\right)x^2-2b(a+c)x+\left(b^2+c^2\right)=0$
This is quadratic in $x$, for equal root $D=0$
$\Rightarrow 4b^2(a+c{)}^2=4\left(a^2+b^2\right)\left(b^2+c^2\right)$
$\Rightarrow b^2{a}^2+b^2{c}^2+2ac{b}^2=a^2{b}^2+a^2{c}^2+b^4+b^2{c}^2$
$\Rightarrow b^4-2ac{b}^2+a^2{c}^2=0$
$\Rightarrow {\left(b^2-ac\right)}^2=0\Rightarrow b^2=ac$
$\therefore a,b,c\text{ are in G.P. }$