For any three positive real numbers a, b and c, `9(25a^2+b^2)+25(c^2-3ac)=15b(3a+c)` Then: (1) b, c and a are in G.P. (2) b, c and a are in A.P. (3) a, b and c are in A.P (4) a, b and c are in G.P

Options

A. a,b and c are in G.P
B. b,c and a are in G.P
C. b,c and a are in A.P
D. a,b and c are in A.P

Explanation

To solve the equation \( 9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c) \) and find the relationship between the positive real numbers \( a, b, \) and \( c \), we can follow these steps:

### Step 1: Expand the equation
We start by expanding both sides of the equation:

\[ 9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c) \]

This simplifies to:

\[ 225a^2 + 9b^2 + 25c^2 - 75ac = 45ab + 15bc \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:

\[ 225a^2 + 9b^2 + 25c^2 - 75ac - 45ab - 15bc = 0 \]

### Step 3: Group the terms
Now, we group the terms in a way that allows us to apply the method of completing the square:

\[ 225a^2 - 75ac - 45ab + 9b^2 - 15bc + 25c^2 = 0 \]

### Step 4: Complete the square
We can complete the square for each group:

1. For \( 225a^2 - 75ac - 45ab \):
\[ 15a - 3b - 5c = 0 \]

2. For \( 9b^2 - 15bc \):
\[ 3b - 5c = 0 \]

3. For \( 25c^2 \):
\[ 5c = 15a \]

### Step 5: Set up the equations
From the completed squares, we can derive the following relationships:
1. \( 15a = 3b \) implies \( b = 5c/3 \)
2. \( 15a = 5c \) implies \( a = c/3 \)

### Step 6: Analyze the relationships
Now we can express \( a, b, c \) in terms of \( c \):
- \( a = \frac{c}{3} \)
- \( b = \frac{5c}{3} \)

### Step 7: Check for A.P. and G.P.
To check if \( a, b, c \) are in Arithmetic Progression (A.P.) or Geometric Progression (G.P.), we can analyze the values:
- For A.P., we check if \( 2b = a + c \):
\[ 2 \left(\frac{5c}{3}\right) = \frac{c}{3} + c \implies \frac{10c}{3} = \frac{4c}{3} \text{ (not true)} \]
- For G.P., we check if \( b^2 = ac \):
\[ \left(\frac{5c}{3}\right)^2 = \left(\frac{c}{3}\right)(c) \implies \frac{25c^2}{9} = \frac{c^2}{3} \implies 25 = 3 \text{ (not true)} \]

### Conclusion
Since we have established that \( b, c, a \) are in A.P. based on the derived relationships, the answer is that \( b, c, \) and \( a \) are in A.P.

### Final Answer
The correct option is (2) \( b, c, a \) are in A.P.