If `|(p,q-y,r-z),(p-x,q,r-z),(p-x,q-y,r)|` = 0 then the value of `p/x+q/y+r/z` is

Options

A. 0
B. 1
C. 2
D. 4 pqr

Answer

To solve the problem, we start with the determinant given as:

\[ D = \begin{vmatrix} p & q-y & r-z \\ p-x & q & r-z \\ p-x & q-y & r \end{vmatrix} \]

We know that if the determinant \( D = 0 \), it indicates that the rows of the determinant are linearly dependent.

### Step 1: Row Operations
We will perform row operations to simplify the determinant. Let's perform the following operations:

- Replace \( R_2 \) with \( R_2 - R_1 \)
- Replace \( R_3 \) with \( R_3 - R_1 \)

After performing these operations, the determinant becomes:

\[ D = \begin{vmatrix} p & q-y & r-z \\ (p-x) - p & q - (q-y) & (r-z) - (r-z) \\ (p-x) - p & (q-y) - (q-y) & r - (r-z) \end{vmatrix} \]

This simplifies to:

\[ D = \begin{vmatrix} p & q-y & r-z \\ -x & y & 0 \\ -x & 0 & z \end{vmatrix} \]

### Step 2: Expanding the Determinant
Now we can expand the determinant using the first row:

\[ D = p \begin{vmatrix} y & 0 \\ 0 & z \end{vmatrix} - (q-y) \begin{vmatrix} -x & 0 \\ -x & z \end{vmatrix} + (r-z) \begin{vmatrix} -x & y \\ -x & 0 \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \( \begin{vmatrix} y & 0 \\ 0 & z \end{vmatrix} = yz \)
2. \( \begin{vmatrix} -x & 0 \\ -x & z \end{vmatrix} = -xz \)
3. \( \begin{vmatrix} -x & y \\ -x & 0 \end{vmatrix} = xy \)

Substituting these back into the determinant gives:

\[ D = p(yz) - (q-y)(-xz) + (r-z)(xy) \]

This simplifies to:

\[ D = pyz + (q-y)xz + (r-z)xy \]

### Step 3: Setting the Determinant to Zero
Since we know \( D = 0 \):

\[ pyz + (q-y)xz + (r-z)xy = 0 \]

### Step 4: Rearranging the Equation
Rearranging the equation gives:

\[ pyz + qxz - yxz + rxy - zxy = 0 \]

Combining like terms:

\[ pyz + qxz + rxy - (yxz + zxy) = 0 \]

### Step 5: Dividing by xyz
To isolate the terms, we divide the entire equation by \( xyz \):

\[ \frac{p}{x} + \frac{q}{y} + \frac{r}{z} - 1 = 0 \]

Thus, we have:

\[ \frac{p}{x} + \frac{q}{y} + \frac{r}{z} = 1 \]

### Final Answer
The value of \( \frac{p}{x} + \frac{q}{y} + \frac{r}{z} \) is:

\[ \boxed{1} \]