If ` (a)/( q - r) = (b)/( r - p) = (c)/(p - q)` find the value of (pa + qb + rc)

Options

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

Answer

To solve the problem, we start with the given equations:

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

where \( k \) is some constant. From this, we can express \( a \), \( b \), and \( c \) in terms of \( k \):

1. **Express \( a \), \( b \), and \( c \)**:
\[ a = k(q - r) \]
\[ b = k(r - p) \]
\[ c = k(p - q) \]

2. **Substitute \( a \), \( b \), and \( c \) into \( pa + qb + rc \)**:
\[ pa + qb + rc = p(k(q - r)) + q(k(r - p)) + r(k(p - q)) \]

3. **Factor out \( k \)**:
\[ = k \left( p(q - r) + q(r - p) + r(p - q) \right) \]

4. **Distribute \( p \), \( q \), and \( r \)**:
\[ = k \left( pq - pr + qr - qp + rp - rq \right) \]

5. **Combine like terms**:
\[ = k \left( pq - qp + qr - rq + rp - pr \right) \]
Notice that \( pq - qp = 0 \), \( qr - rq = 0 \), and \( rp - pr = 0 \). Thus, all terms cancel out:
\[ = k(0) = 0 \]

6. **Final Result**:
\[ pa + qb + rc = 0 \]