If one root of the equation `ax^(2)+bx+c=0` be reciprocal of the one root of the equation `d' x^(2)+b'x+c'=0` then the required condition is …..

Answer

To solve the problem, we need to establish the condition under which one root of the equation \( ax^2 + bx + c = 0 \) is the reciprocal of one root of the equation \( d'x^2 + b'x + c' = 0 \).

### Step-by-Step Solution:

1. **Identify the Common Root**:
Let \( \alpha \) be the common root of both equations. According to the problem, one root of the first equation is \( \alpha \) and the reciprocal of one root of the second equation is \( \frac{1}{\alpha} \).

2. **Set Up the Equations**:
Since \( \alpha \) is a root of the first equation, we have:
\[ a\alpha^2 + b\alpha + c = 0 \quad \text{(1)} \]
For the second equation, since \( \frac{1}{\alpha} \) is a root, substituting \( x = \frac{1}{\alpha} \) gives:
\[ d'\left(\frac{1}{\alpha}\right)^2 + b'\left(\frac{1}{\alpha}\right) + c' = 0 \]
Multiplying through by \( \alpha^2 \) to eliminate the fraction:
\[ d' + b'\alpha + c'\alpha^2 = 0 \quad \text{(2)} \]

3. **Rearranging the Equations**:
From equation (1):
\[ a\alpha^2 + b\alpha + c = 0 \implies a\alpha^2 + b\alpha = -c \]
From equation (2):
\[ c'\alpha^2 + b'\alpha + d' = 0 \implies c'\alpha^2 + b'\alpha = -d' \]

4. **Cross-Multiplication**:
We can set up a relationship between the coefficients from both equations. We will use the method of cross-multiplication to derive the required condition:
\[ \frac{b\alpha + c}{a\alpha^2} = \frac{b'\alpha + d'}{c'\alpha^2} \]

5. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ (b\alpha + c)(c'\alpha^2) = (b'\alpha + d')(a\alpha^2) \]

6. **Expanding Both Sides**:
Expanding both sides leads to:
\[ bc'\alpha^2 + cc' = ab'\alpha^2 + ad' \]

7. **Rearranging Terms**:
Rearranging the equation gives:
\[ (bc' - ab')\alpha^2 + (c - ad') = 0 \]

8. **Setting the Coefficient of \( \alpha^2 \) to Zero**:
For this equation to hold for all \( \alpha \), both coefficients must be zero:
\[ bc' - ab' = 0 \quad \text{and} \quad c - ad' = 0 \]

9. **Final Required Condition**:
Thus, the required condition is:
\[ \frac{b}{b'} = \frac{c}{c'} = \frac{a}{d'} \]