Question

If $p,q,r$ are in G.P and the equations, $p{x}^2+2qx+r=0$ and $d{x}^2+2ex+f=0$ have a common root, then show that $\frac{d}{p},\frac{e}{q},\frac{f}{r}$ are in an A.P.

Answer 1

$p,q,r\to G.P$

$\Rightarrow q^2=pr$ .... (1)

$p{x}^2+2qx+r=0$

$d{x}^2+2ex+f=0$ common roots

Condn of common root : -

$(rd–Pf{)}^2=(4pe–4qd\left)\right(4qf–4er)$

$\Rightarrow r^2{d}^2+p^2{f}^2–2rdpf=16[peqf–pr{e}^2–q^{2}df–qder]$

$\Rightarrow {(\frac{2e}{q}-\frac{d}{p}-\frac{f}{r})}^2=0$

$\Rightarrow \frac{2e}{q}=\frac{d}{p}+\frac{f}{r}$

Hence proved.