Question

If the roots of $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0$ are real and equal then $a,b,c$ are in

• A. $A.P$
• B. $H.P$
• C. $G.P$
• D. $A.G.P$

Answer 1

The correct option is C $G.P$

Given equation $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0$ roots are real and equal $\Rightarrow$ $D=0$

${[-2b(a+c\left)\right]}^2-4(a^2+b^2)(b^2+c^2)=0$

$4b^2{(a+c)}^2-4(a^2+b^2)(b^2+c^2)=0$

$\Rightarrow$ $b^2{a}^2+b^2{c}^2+2b^{2}ac=a^2{b}^2+a^2{c}^2+b^4+b^2{c}^2$

$\Rightarrow$ ${(ac-b^2)}^2=0$

$\Rightarrow$ $b^2=ac$

$\Rightarrow$ $a,b,c$ are in G.P