`a ,b ,c` are positive real numbers forming a G.P. If `ax^2+2b x+c=0 and dx^2+2e x+f=0` have a common root, then prove that `d//a ,e//b ,f//c` are in A.P.

Explanation

To prove that \( \frac{d}{a}, \frac{e}{b}, \frac{f}{c} \) are in arithmetic progression (A.P.), given that \( a, b, c \) are positive real numbers forming a geometric progression (G.P.) and the equations \( ax^2 + 2bx + c = 0 \) and \( dx^2 + 2ex + f = 0 \) have a common root, we can follow these steps:

### Step 1: Establish the relationship from G.P.
Since \( a, b, c \) are in G.P., we have:
\[ \frac{b}{a} = \frac{c}{b} \]
This implies:
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