If the system of linear equations.
`x +4ay+az=0`
`x+ 3by +bz=0`
`x+2cy + cz=0`
have a non-zero solution, then a,b,c are in .

Options

A. A.P.
B. G.P.
C. H.P.
D. None

Answer

To determine the relationship between \( a, b, c \) such that the system of linear equations has a non-zero solution, we need to analyze the determinant of the coefficient matrix.

The given equations are:

1. \( x + 4ay + az = 0 \)
2. \( x + 3by + bz = 0 \)
3. \( x + 2cy + cz = 0 \)

### Step 1: Form the Coefficient Matrix
The coefficient matrix \( A \) of the system can be written as:

\[ A = \begin{bmatrix} 1 & 4a & a \\ 1 & 3b & b \\ 1 & 2c & c \end{bmatrix} \]

### Step 2: Calculate the Determinant
To find the condition for a non-trivial solution, we need to calculate the determinant of matrix \( A \) and set it equal to zero:

\[ \text{det}(A) = \begin{vmatrix} 1 & 4a & a \\ 1 & 3b & b \\ 1 & 2c & c \end{vmatrix} \]

Using the determinant formula for a 3x3 matrix, we expand it:

\[ \text{det}(A) = 1 \cdot \begin{vmatrix} 3b & b \\ 2c & c \end{vmatrix} - 4a \cdot \begin{vmatrix} 1 & b \\ 1 & c \end{vmatrix} + a \cdot \begin{vmatrix} 1 & 3b \\ 1 & 2c \end{vmatrix} \]

### Step 3: Calculate the 2x2 Determinants
Now we calculate the 2x2 determinants:

1. \( \begin{vmatrix} 3b & b \\ 2c & c \end{vmatrix} = 3bc - 2bc = bc \)

2. \( \begin{vmatrix} 1 & b \\ 1 & c \end{vmatrix} = c - b \)

3. \( \begin{vmatrix} 1 & 3b \\ 1 & 2c \end{vmatrix} = 2c - 3b \)

### Step 4: Substitute Back into the Determinant
Substituting these back into the determinant expression:

\[ \text{det}(A) = 1 \cdot (bc) - 4a(c - b) + a(2c - 3b) \]

Expanding this gives:

\[ \text{det}(A) = bc - 4ac + 4ab + 2ac - 3ab \]

Combining like terms:

\[ \text{det}(A) = bc + (4ab - 3ab) + (-4ac + 2ac) = bc + ab - 2ac \]

### Step 5: Set the Determinant to Zero
For a non-trivial solution, we set the determinant equal to zero:

\[ bc + ab - 2ac = 0 \]

### Step 6: Rearranging the Equation
Rearranging gives:

\[ ab + bc = 2ac \]

### Step 7: Expressing the Relationship
This can be expressed as:

\[ \frac{a}{c} + \frac{b}{c} = 2 \]

This implies that \( a, b, c \) are in Harmonic Progression (HP).

### Conclusion
Thus, the values of \( a, b, c \) must be in Harmonic Progression.

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