If p ad q are non-zero constants, the equation `x^(2)+px+q=0` has roots `alpha and beta`, then the equation `qx^(2)+px+1=0` has roots

Options

A. `alpha and 1//beta`
B. `1//alpha and beta`
C. `1//alpha and 1//beta`
D. none of these

Answer

To solve the problem, we need to analyze the two quadratic equations given and their roots. Let's break it down step by step.

### Step 1: Identify the roots of the first equation
The first equation is given as:
\[ x^2 + px + q = 0 \]
Let the roots of this equation be \( \alpha \) and \( \beta \).

Using Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{p}{1} = -p \)
- The product of the roots \( \alpha \beta = \frac{q}{1} = q \)

### Step 2: Analyze the second equation
The second equation is:
\[ qx^2 + px + 1 = 0 \]
We need to find the roots of this equation, which we will denote as \( \alpha_1 \) and \( \beta_1 \).

Again, applying Vieta's formulas for this equation:
- The sum of the roots \( \alpha_1 + \beta_1 = -\frac{p}{q} \)
- The product of the roots \( \alpha_1 \beta_1 = \frac{1}{q} \)

### Step 3: Relate the roots of the second equation to the first
We need to find roots \( \alpha_1 \) and \( \beta_1 \) such that:
1. Their sum \( \alpha_1 + \beta_1 = -\frac{p}{q} \)
2. Their product \( \alpha_1 \beta_1 = \frac{1}{q} \)

### Step 4: Check possible forms for \( \alpha_1 \) and \( \beta_1 \)
We can express the roots \( \alpha_1 \) and \( \beta_1 \) in terms of \( \alpha \) and \( \beta \). A common transformation is to take the reciprocals of the roots of the first equation:
\[ \alpha_1 = \frac{1}{\alpha}, \quad \beta_1 = \frac{1}{\beta} \]

### Step 5: Verify the sum and product
Now, we calculate the sum and product of \( \alpha_1 \) and \( \beta_1 \):
- The sum:
\[ \alpha_1 + \beta_1 = \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} = \frac{-p}{q} \]
- The product:
\[ \alpha_1 \beta_1 = \frac{1}{\alpha} \cdot \frac{1}{\beta} = \frac{1}{\alpha \beta} = \frac{1}{q} \]

Both conditions are satisfied.

### Conclusion
Thus, the roots \( \alpha_1 \) and \( \beta_1 \) of the equation \( qx^2 + px + 1 = 0 \) are:
\[ \alpha_1 = \frac{1}{\alpha}, \quad \beta_1 = \frac{1}{\beta} \]