If the roots of the equation `x^(2) + px + q = 0` are in the same ratio as those of the equation `x^(2) + lx + m = 0`. Then which one of the following is correct ?

Options

A. a) `p^(2) m = l^(2) q`
B. b) `m^(2) p = l^(2) q`
C. c) `m^(2) p = q^(2) l`
D. d) `m^(2) p^(2) = l^(2) q`

Answer

To solve the problem, we need to establish the relationship between the roots of the two quadratic equations given. Let's break it down step by step.

### Step 1: Identify the roots of the equations
We have two quadratic equations:
1. \( x^2 + px + q = 0 \)
2. \( x^2 + lx + m = 0 \)

Let the roots of the first equation be \( \alpha \) and \( \beta \), and the roots of the second equation be \( \gamma \) and \( \delta \).

### Step 2: Use Vieta's formulas
According to Vieta's formulas:
- For the first equation:
- Sum of roots: \( \alpha + \beta = -p \)
- Product of roots: \( \alpha \beta = q \)

- For the second equation:
- Sum of roots: \( \gamma + \delta = -l \)
- Product of roots: \( \gamma \delta = m \)

### Step 3: Establish the ratio of the roots
We are given that the roots of the first equation are in the same ratio as those of the second equation. This can be expressed as:
\[ \frac{\alpha}{\beta} = \frac{\gamma}{\delta} \]
This implies that:
\[ \alpha \delta = \beta \gamma \]

### Step 4: Express the sums and products in terms of \( \alpha \) and \( \beta \)
From the ratio \( \frac{\alpha}{\beta} = k \) (for some constant \( k \)), we can express:
- \( \alpha = k\beta \)

Substituting this into Vieta's formulas:
- \( \alpha + \beta = k\beta + \beta = (k + 1)\beta = -p \)
- Thus, \( \beta = \frac{-p}{k + 1} \)
- Therefore, \( \alpha = k\beta = \frac{-kp}{k + 1} \)

### Step 5: Calculate the product of the roots
Now, using the product of the roots:
\[ \alpha \beta = q \]
Substituting for \( \alpha \) and \( \beta \):
\[ \left(\frac{-kp}{k + 1}\right)\left(\frac{-p}{k + 1}\right) = q \]
This simplifies to:
\[ \frac{k p^2}{(k + 1)^2} = q \]

### Step 6: Repeat for the second equation
Similarly, for the second equation, we can express:
- \( \gamma = k\delta \)

Using Vieta's formulas:
- \( \gamma + \delta = k\delta + \delta = (k + 1)\delta = -l \)
- Thus, \( \delta = \frac{-l}{k + 1} \)
- Therefore, \( \gamma = k\delta = \frac{-kl}{k + 1} \)

Calculating the product:
\[ \gamma \delta = m \]
Substituting for \( \gamma \) and \( \delta \):
\[ \left(\frac{-kl}{k + 1}\right)\left(\frac{-l}{k + 1}\right) = m \]
This simplifies to:
\[ \frac{k l^2}{(k + 1)^2} = m \]

### Step 7: Establish the relationship between \( p, q, l, m \)
From the expressions we derived:
1. \( q = \frac{k p^2}{(k + 1)^2} \)
2. \( m = \frac{k l^2}{(k + 1)^2} \)

Setting these equal gives:
\[ \frac{p^2}{q} = \frac{l^2}{m} \]
Cross-multiplying yields:
\[ p^2 m = l^2 q \]

### Conclusion
Thus, the correct relationship is:
\[ p^2 m = l^2 q \]