If `p q r!=0` and the system of equation `(p+a)x+b y+c z=0` `a x+(q+b)y+c z=0` `a c+b y+(r+c)z=0` has nontrivial solution, then value of `a/p+b/q+c/r` is `-1` b. `0` c.`0""` d. `not-2`

Options

A. -1
B. 0
C. 1
D. 2

Explanation

To solve the given system of equations for the condition of nontrivial solutions, we will use the determinant of the coefficients matrix. The equations are:

1. \((p + a)x + by + cz = 0\)
2. \(ax + (q + b)y + cz = 0\)
3. \(a x + by + (r + c)z = 0\)

### Step 1: Form the Coefficient Matrix
We can represent the system of equations in matrix form as follows:

\[ \begin{bmatrix} p + a & b & c \\ a & q + b & c \\ a & b & r + c \end{bmatrix} \]

### Step 2: Set the Determinant to Zero
For the system to have a nontrivial solution, the determinant of the coefficient matrix must be zero:

\[ \text{det} \begin{bmatrix} p + a & b & c \\ a & q + b & c \\ a & b & r + c \end{bmatrix} = 0 \]

### Step 3: Calculate the Determinant
We will calculate the determinant using the cofactor expansion along the first row:

\[ \text{det} = (p + a) \begin{vmatrix} q + b & c \\ b & r + c \end{vmatrix} - b \begin{vmatrix} a & c \\ a & r + c \end{vmatrix} + c \begin{vmatrix} a & q + b \\ a & b \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \(\begin{vmatrix} q + b & c \\ b & r + c \end{vmatrix} = (q + b)(r + c) - bc\)

2. \(\begin{vmatrix} a & c \\ a & r + c \end{vmatrix} = a(r + c) - ac = ar\)

3. \(\begin{vmatrix} a & q + b \\ a & b \end{vmatrix} = ab - a(q + b) = a(b - q - b) = -aq\)

Substituting these back into the determinant:

\[ \text{det} = (p + a)((q + b)(r + c) - bc) - b(ar) + c(-aq) \]

### Step 4: Simplify the Determinant
Setting the determinant equal to zero gives us:

\[ (p + a)((q + b)(r + c) - bc) - bar - acq = 0 \]

### Step 5: Rearranging the Equation
After simplifying, we can express the equation in terms of \(a/p + b/q + c/r\):

\[ \frac{a}{p} + \frac{b}{q} + \frac{c}{r} = -1 \]

### Conclusion
Thus, the value of \(\frac{a}{p} + \frac{b}{q} + \frac{c}{r}\) is \(-1\).

### Final Answer
The correct option is:
**a. -1**

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