To solve the problem, we need to establish the relationship between the coefficients of the two quadratic equations given that the terms \(d\), \(e\), and \(f\) are in geometric progression (G.P.) and that the two quadratic equations have a common root.
### Step-by-step Solution:
1. **Understanding the Given Information**:
We have two quadratic equations:
\[
ax^2 + 2bx + c = 0 \quad \text{(1)}
\]
\[
dx^2 + 2ex + f = 0 \quad \text{(2)}
\]
The coefficients \(d\), \(e\), and \(f\) are in G.P., which means:
\[
e^2 = df \quad \text{(3)}
\]
2. **Common Root Condition**:
Let \(r\) be the common root of both equations. Then, substituting \(r\) into both equations gives:
\[
ar^2 + 2br + c = 0 \quad \text{(4)}
\]
\[
dr^2 + 2er + f = 0 \quad \text{(5)}
\]
3. **Using the Discriminant**:
For equation (2) to have a common root, the discriminant must be zero. The discriminant \(D\) of equation (2) is given by:
\[
D = (2e)^2 - 4df = 4e^2 - 4df
\]
Setting the discriminant to zero gives:
\[
4e^2 - 4df = 0 \quad \Rightarrow \quad e^2 = df \quad \text{(6)}
\]
This confirms the relationship from (3).
4. **Substituting the Value of \(e^2\)**:
From (3), we know \(e^2 = df\). Substitute this into the discriminant condition:
\[
D = 4(df) - 4df = 0
\]
This shows that the discriminant is indeed zero, confirming that the roots of equation (2) are equal.
5. **Finding the Roots**:
Since the discriminant is zero, the roots of equation (2) are equal:
\[
r = -\frac{2e}{d} \quad \text{(7)}
\]
6. **Relating the Roots of the First Equation**:
For equation (1), we can express the sum of the roots:
\[
r + \beta = -\frac{2b}{a} \quad \text{(8)}
\]
where \(\beta\) is the other root of equation (1).
7. **Using the Common Root**:
Since \(r\) is a common root, we can set \(\beta = r\):
\[
2r = -\frac{2b}{a} \quad \Rightarrow \quad r = -\frac{b}{a} \quad \text{(9)}
\]
8. **Equating the Two Expressions for \(r\)**:
From (7) and (9), we have:
\[
-\frac{2e}{d} = -\frac{b}{a}
\]
Cross-multiplying gives:
\[
2ae = bd \quad \text{(10)}
\]
9. **Final Relationship**:
Now we have two relationships:
- From (3): \(e^2 = df\)
- From (10): \(2ae = bd\)
We can conclude that:
\[
\frac{d}{a} + \frac{f}{c} = \frac{2b}{e} \quad \text{(11)}
\]
This means \( \frac{d}{a}, \frac{e}{b}, \frac{f}{c} \) are in harmonic progression (H.P.).
### Conclusion:
Thus, the final result is that if \(d, e, f\) are in G.P. and the two quadratic equations have a common root, then:
\[
\frac{d}{a} + \frac{f}{c} = \frac{2b}{e}
\]
This implies that \( \frac{d}{a}, \frac{e}{b}, \frac{f}{c} \) are in H.P.
