If `alpha, beta` be the roots of the equation `ax^(2)+bx+c=0`, then the roots of the equation `ax^(2)+blambda x+c lambda^(2)=0`, `lambda^(2) !=0`, are

Options

A. `lambda alpha, lambda beta`
B. `(alpha)/(lambda), (beta)/(lambda)`
C. `alpha, beta`
D. `alpha^(k), beta^(k)`

Answer

To find the roots of the equation \( ax^2 + b\lambda x + c\lambda^2 = 0 \), given that \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the roots of the original equation**:
The roots \( \alpha \) and \( \beta \) of the equation \( ax^2 + bx + c = 0 \) can be expressed using Vieta's formulas:
\[ \alpha + \beta = -\frac{b}{a} \quad \text{(sum of the roots)} \]
\[ \alpha \beta = \frac{c}{a} \quad \text{(product of the roots)} \]

2. **Rewrite the new equation**:
We need to analyze the new equation:
\[ ax^2 + b\lambda x + c\lambda^2 = 0 \]
We can rearrange this equation by dividing through by \( a \):
\[ x^2 + \frac{b\lambda}{a} x + \frac{c\lambda^2}{a} = 0 \]

3. **Express the coefficients in terms of \( \alpha \) and \( \beta \)**:
From Vieta's formulas, we can express the new coefficients in terms of \( \alpha \) and \( \beta \):
- The sum of the roots of the new equation is:
\[ -\frac{b\lambda}{a} = -\lambda(\alpha + \beta) = -\lambda\left(-\frac{b}{a}\right) = \frac{b\lambda}{a} \]
- The product of the roots of the new equation is:
\[ \frac{c\lambda^2}{a} = \lambda^2(\alpha \beta) = \lambda^2\left(\frac{c}{a}\right) \]

4. **Identify the new roots**:
From the above expressions, we can see that the roots of the new equation can be written as:
\[ \text{Roots} = \lambda \alpha \quad \text{and} \quad \lambda \beta \]

5. **Conclusion**:
Therefore, the roots of the equation \( ax^2 + b\lambda x + c\lambda^2 = 0 \) are:
\[ \lambda \alpha \quad \text{and} \quad \lambda \beta \]