If ` a^(2) + 13 b^(2) + c^(2) - 4 ab - 6 b c = 0 ` then a : b : c is

Options

A. A)` 1 : 2 : 3 `
B. B)` 2 : 3 : 1 `
C. C)` 2 : 1 : 3 `
D. D)` 1 : 3 : 2 `

Answer

To solve the equation \( a^2 + 13b^2 + c^2 - 4ab - 6bc = 0 \) and find the ratio \( a : b : c \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ a^2 + 13b^2 + c^2 - 4ab - 6bc = 0 \]
We can rearrange the terms to group them in a way that might help us complete the square.

### Step 2: Completing the Square for \( a \)
We can rewrite the terms involving \( a \) and \( b \):
\[ a^2 - 4ab + 13b^2 + c^2 - 6bc = 0 \]
Now, we complete the square for \( a \) and \( b \):
\[ (a - 2b)^2 - 4b^2 + 13b^2 + c^2 - 6bc = 0 \]
This simplifies to:
\[ (a - 2b)^2 + 9b^2 + c^2 - 6bc = 0 \]

### Step 3: Completing the Square for \( b \) and \( c \)
Next, we complete the square for the terms involving \( b \) and \( c \):
\[ (a - 2b)^2 + (c - 3b)^2 = 0 \]
This means both squares must equal zero:
\[ a - 2b = 0 \quad \text{and} \quad c - 3b = 0 \]

### Step 4: Solving the Equations
From the equations:
1. \( a - 2b = 0 \) implies \( a = 2b \)
2. \( c - 3b = 0 \) implies \( c = 3b \)

### Step 5: Finding the Ratio
Now we can express \( a \), \( b \), and \( c \) in terms of \( b \):
\[ a = 2b, \quad b = b, \quad c = 3b \]
Thus, the ratio \( a : b : c \) is:
\[ 2b : b : 3b \]
This simplifies to:
\[ 2 : 1 : 3 \]

### Final Answer
The ratio \( a : b : c \) is \( 2 : 1 : 3 \).

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