Question

Let $a,b,c$ are in G.P. and the equations $a{x}^2+2bx+c=0$ and $p{x}^2+2qx+r=0$ have one common root $(a,b,c,p,q,r\in R,a,p\ne 0).$ Then

• A. $ar,bq,cp$ are in G.P.
• B. $ar,bq,cp$ are in H.P.
• C. $ar,bq,cp$ are in A.P.
• D. $bq,ar,cp$ are in A.P.

Answer 1

The correct option is C $ar,bq,cp$ are in A.P.

Given that $b^2=ac$
For $a{x}^2+2bx+c=0$
$\Delta =4b^2-4ac=0$
So, the equation has equal roots. Let that root be $\alpha .$
Then, $\alpha =-\frac{b}{a}$

It is a root of the equation $p{x}^2+2qx+r=0$
$\Rightarrow p{(-\frac{b}{a})}^2+2q(-\frac{b}{a})+r=0\Rightarrow b^{2}p-2abq+r{a}^2=0\Rightarrow ra+cp=2bq$