The lines `px +qy+r=0, qx + ry + p =0,rx + py+q=0,` are concurrant then

Options

A. `p+q+r=pqr`
B. `p^(3)+q^(3)+r^(3)=3pqr`
C. `p^(2)+q^(2)+r^(2)=2(pq+qr+rp)`
D. None of these

Answer

To determine the condition for the lines \( px + qy + r = 0 \), \( qx + ry + p = 0 \), and \( rx + py + q = 0 \) to be concurrent, we will use the concept of determinants.

### Step-by-Step Solution:

1. **Identify the coefficients**:
For the lines given, we can identify the coefficients as follows:
- For the first line \( px + qy + r = 0 \):
- \( a_1 = p \), \( b_1 = q \), \( c_1 = r \)
- For the second line \( qx + ry + p = 0 \):
- \( a_2 = q \), \( b_2 = r \), \( c_2 = p \)
- For the third line \( rx + py + q = 0 \):
- \( a_3 = r \), \( b_3 = p \), \( c_3 = q \)

2. **Set up the determinant**:
The lines are concurrent if the determinant of the matrix formed by the coefficients is zero:
\[ \begin{vmatrix} p & q & r \\ q & r & p \\ r & p & q \end{vmatrix} = 0 \]

3. **Calculate the determinant**:
We can calculate the determinant using the formula for a 3x3 matrix:
\[ D = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2) \]
Substituting the values:
\[ D = p(rq - pr) - q(qp - rp) + r(qp - rq) \]

4. **Simplify the expression**:
Expanding the determinant:
\[ D = prq - p^2r - q^2p + qrp + rq^2 - r^2q \]
Rearranging gives:
\[ D = pqr - p^2r - q^2p + qrp + rq^2 - r^2q \]

5. **Set the determinant to zero**:
For the lines to be concurrent:
\[ pqr - p^2r - q^2p + qrp + rq^2 - r^2q = 0 \]

6. **Factor and simplify**:
We can factor the expression:
\[ pqr + rqp - p^2r - q^2p - r^2q = 0 \]
This can be rearranged to find relationships among \( p, q, r \).

7. **Conclusion**:
After simplification, we can derive the relationship that must hold true for the lines to be concurrent. The final condition can be expressed as:
\[ p^2 + q^2 + r^2 = pq + qr + rp \]