Let three quadratic equations ` ax^(2) - 2bx + c = 0, bx^(2) - 2 cx + a = 0`
and `cx^(2) -2 ax + b = 0 `, all have only positive roots. Then ltbr. Which of these are always ture?

Options

A. `b^(2) = ac `
B. `c^(2) = ab `
C. each pair of equations has exactly one root common
D. each pair of equations has two roots common

Explanation

To solve the problem, we need to analyze the three quadratic equations given and apply the conditions for them to have only positive roots. The equations are:

1. \( ax^2 - 2bx + c = 0 \)
2. \( bx^2 - 2cx + a = 0 \)
3. \( cx^2 - 2ax + b = 0 \)

### Step 1: Conditions for Positive Roots
For a quadratic equation \( Ax^2 + Bx + C = 0 \) to have positive roots, the following conditions must be satisfied:

1. The discriminant \( D \) must be non-negative: \( D = B^2 - 4AC \geq 0 \).
2. The leading coefficient \( A \) must be positive: \( A > 0 \).
3. The constant term \( C \) must be positive: \( C > 0 \).

### Step 2: Apply Conditions to Each Equation
Let's apply these conditions to each of the three equations.

**For the first equation \( ax^2 - 2bx + c = 0 \):**
- Discriminant: \( (-2b)^2 - 4ac \geq 0 \) implies \( 4b^2 \geq 4ac \) or \( b^2 \geq ac \).
- Leading coefficient: \( a > 0 \).
- Constant term: \( c > 0 \).

**For the second equation \( bx^2 - 2cx + a = 0 \):**
- Discriminant: \( (-2c)^2 - 4ab \geq 0 \) implies \( 4c^2 \geq 4ab \) or \( c^2 \geq ab \).
- Leading coefficient: \( b > 0 \).
- Constant term: \( a > 0 \).

**For the third equation \( cx^2 - 2ax + b = 0 \):**
- Discriminant: \( (-2a)^2 - 4bc \geq 0 \) implies \( 4a^2 \geq 4bc \) or \( a^2 \geq bc \).
- Leading coefficient: \( c > 0 \).
- Constant term: \( b > 0 \).

### Step 3: Summary of Conditions
From the above analysis, we have the following inequalities:
1. \( b^2 \geq ac \)
2. \( c^2 \geq ab \)
3. \( a^2 \geq bc \)

### Step 4: Multiplying the Inequalities
Now, if we multiply these inequalities together, we get:
\[ (b^2)(c^2)(a^2) \geq (ac)(ab)(bc) \]
This simplifies to:
\[ b^2c^2a^2 \geq a^2b^2c^2 \]
This implies that the conditions are consistent and can lead us to conclude that:
\[ a^2 = bc, \quad b^2 = ac, \quad c^2 = ab \]

### Step 5: Conclusion
From the analysis, we can conclude that:
- Each of the conditions \( b^2 = ac \), \( c^2 = ab \), and \( a^2 = bc \) holds true.
- Therefore, the options that are always true are:
- Option A: \( b^2 = ac \)
- Option B: \( c^2 = ab \)
- Option D: Each pair of equations has exactly one common root.