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

Options

A. `A.P.`
B. `G.P.`
C. `H.P.`
D. satisfies `a+2b+3c=0`

Explanation

To solve the problem, we need to determine the relationship between \( a, b, c \) given that the system of linear equations has a non-zero solution. The equations are:

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

### Step 1: Set up the determinant

For the system to have a non-zero solution, the determinant of the coefficients must be zero. The coefficients of the equations can be arranged into a matrix:

\[ \begin{vmatrix} 1 & 2a & a \\ 1 & 3b & b \\ 1 & 4c & c \end{vmatrix} \]

### Step 2: Calculate the determinant

Using the determinant formula for a 3x3 matrix, we can expand the determinant as follows:

\[ \Delta = 1 \cdot \begin{vmatrix} 3b & b \\ 4c & c \end{vmatrix} - 2a \cdot \begin{vmatrix} 1 & b \\ 1 & c \end{vmatrix} + a \cdot \begin{vmatrix} 1 & 3b \\ 1 & 4c \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} 3b & b \\ 4c & c \end{vmatrix} = 3bc - 4bc = -bc \)

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

3. \( \begin{vmatrix} 1 & 3b \\ 1 & 4c \end{vmatrix} = 4c - 3b \)

Substituting these back into the determinant:

\[ \Delta = 1(-bc) - 2a(c - b) + a(4c - 3b) \]

### Step 3: Simplify the determinant

Now, we simplify:

\[ \Delta = -bc - 2a(c - b) + a(4c - 3b) \]
\[ = -bc - 2ac + 2ab + 4ac - 3ab \]
\[ = (-bc + 2ab + 2ac - 3ab) = -bc + 2ac - ab \]

### Step 4: Set the determinant to zero

For the system to have a non-zero solution, we set the determinant to zero:

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

### Step 5: Rearranging the equation

Rearranging gives:

\[ 2ac = ab + bc \]

### Step 6: Divide by abc (assuming \( a, b, c \neq 0 \))

Dividing through by \( abc \):

\[ \frac{2ac}{abc} = \frac{ab}{abc} + \frac{bc}{abc} \]
\[ \frac{2}{b} = \frac{1}{c} + \frac{1}{a} \]

### Step 7: Recognizing the relationship

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

### Conclusion

Thus, the relationship between \( a, b, c \) is that they are in HP.

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