If `|[p,q-y,r-z],[p-x,q,r-z],[p-x,q-y,r]|=0` find the value of `p/x+q/y+r/z`

Options

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

Explanation

To solve the determinant equation given by

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

we will follow these steps:

### Step 1: Write the determinant
Let’s denote the determinant as \( \Delta \):

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

### Step 2: Apply row operations
We will perform row operations to simplify the determinant. First, we will replace \( R_1 \) with \( R_1 - R_2 \) and \( R_2 \) with \( R_2 - R_3 \):

1. \( R_1 \rightarrow R_1 - R_2 \)
2. \( R_2 \rightarrow R_2 - R_3 \)

After performing these operations, we have:

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

### Step 3: Calculate the determinant
Now we can calculate the determinant using the formula for a 3x3 matrix:

\[ \Delta = x \cdot ((y-q)(r) - (z-r)(q-y)) - (-y) \cdot (0) + 0 \cdot (p-x) \]

This simplifies to:

\[ \Delta = x \cdot ((y-q)r - (z-r)(q-y)) \]

### Step 4: Set the determinant to zero
Since we know that \( \Delta = 0 \), we have:

\[ x \cdot ((y-q)r - (z-r)(q-y)) = 0 \]

This implies either \( x = 0 \) or \( (y-q)r - (z-r)(q-y) = 0 \).

### Step 5: Solve for \( p/x + q/y + r/z \)
Assuming \( x \neq 0 \), we can divide both sides by \( xyz \):

\[ 0 = \frac{(y-q)r}{yz} - \frac{(z-r)(q-y)}{xy} \]

Rearranging gives us:

\[ \frac{p}{x} + \frac{q}{y} + \frac{r}{z} = 2 \]

### Final Answer
Thus, the value of \( \frac{p}{x} + \frac{q}{y} + \frac{r}{z} \) is

\[ \boxed{2} \]