If `p, q, r` each are positive rational number such tlaht `p gt q gt r` and the quadratic equation `(p + q - 2r)x^(2) + (q + r- 2p)x + (r + p - 2q) = 0` has a root in `(-1 , 0)` then which of the following statement hold good? (A) `(r + p)/(q) lt 2` (B) Both roots of given quadratic are rational (C) The equation `px^(2) + 2qx + r = 0` has real and distinct roots (D) The equation `px^(2) + 2qx + r = 0` has no real roots

Options

A. `(r + p)/(q) lt 2`
B. Both roots of given quadratic are rational
C. The equation `px^(2) + 2qx + r = 0` has real and distinct roots
D. The equation `px^(2) + 2qx + r = 0` has no real roots

Explanation

To solve the problem, we need to analyze the given quadratic equation and the conditions provided.

The quadratic equation is given as:
\[ (p + q - 2r)x^2 + (q + r - 2p)x + (r + p - 2q) = 0 \]

### Step 1: Identify the coefficients
Let:
- \( A = p + q - 2r \)
- \( B = q + r - 2p \)
- \( C = r + p - 2q \)

### Step 2: Check if the quadratic has a root in the interval (-1, 0)
We need to check the value of the quadratic equation at \( x = -1 \) and \( x = 0 \).

1. **Calculate \( f(-1) \)**:
\[ f(-1) = A(-1)^2 + B(-1) + C = A - B + C \]
\[ = (p + q - 2r) - (q + r - 2p) + (r + p - 2q) \]
Simplifying this:
\[ = p + q - 2r - q - r + 2p + r + p - 2q \]
\[ = 4p - 2q - 2r \]

2. **Calculate \( f(0) \)**:
\[ f(0) = C = r + p - 2q \]

### Step 3: Determine the conditions for a root in (-1, 0)
For the quadratic to have a root in the interval (-1, 0), we need:
\[ f(-1) \cdot f(0) < 0 \]
This means:
\[ (4p - 2q - 2r)(r + p - 2q) < 0 \]

### Step 4: Analyze the options
Now, we will analyze the given options based on the conditions derived.

#### Option (A): \(\frac{r + p}{q} < 2\)
Rearranging gives:
\[ r + p < 2q \]
This can be derived from the conditions we have established.

#### Option (B): Both roots of the given quadratic are rational
Since \( p, q, r \) are rational, and the coefficients are rational, the roots will also be rational if the discriminant is a perfect square.

#### Option (C): The equation \( px^2 + 2qx + r = 0 \) has real and distinct roots
The discriminant for this equation is:
\[ D = (2q)^2 - 4pr = 4q^2 - 4pr \]
For this to be positive, we need \( q^2 > pr \).

#### Option (D): The equation \( px^2 + 2qx + r = 0 \) has no real roots
This is the opposite of option C, and thus cannot hold if C is true.

### Conclusion
Based on the analysis:
- **Option (A)** is likely true.
- **Option (B)** is true as both roots are rational.
- **Option (C)** is true if the discriminant is positive.
- **Option (D)** is false if C is true.

Thus, the correct options are **A, B, and C**.