Question

If $a , b , c$ are in G.P., then the equations, $ax ^2+2 bx + c =0$ and $dx x ^2+2 ex + f =0$ have $a$ common root, if $\frac{ d }{ a }, \frac{ e }{ b }, \frac{ f }{ c }$ are in :

• A. A.P.
• B. G.P.
• C. H.P.
• D. none

Answer 1

Answer: (A)

$ a, b, c$ are in G.P. $\Rightarrow b^2=a c$
Discriminant of $a x^2+2 b x+c=0$ is $4 b^2-4 a c=4 a c-4 a c=0$
$\Rightarrow $ roots are coincident and equals $-\frac{2 b }{2 a }=-\frac{ b }{ a }$
This must be the common root. Hence $x =-\frac{ b }{ a }$ must satisfy $dx x ^2+2 ex + f =0$
$\therefore d \frac{ b ^2}{ a ^2}-2 e \frac{ b }{ a }+ f =0 \Rightarrow db ^2-2 eba + fa ^2=0 \Rightarrow dac -2 eba + fa ^2=0$
dividing by $a ^2 c$,
$\frac{ d }{ a }-\frac{2 eb }{ ac }+\frac{ f }{ c }=0 $
$\frac{ d }{ a }-\frac{2 e }{ b }+\frac{ f }{ c }=0 \left( b ^2= ac \right)$
$\Rightarrow \frac{ d }{ a }, \frac{ e }{ b }, \frac{ f }{ c } \operatorname{are~in} A . P$