Question

The lines \(px+qy+r=0\), \(qx+ry+p=0\) and \(rx+py+q=0\) are concurrent if

• A. \(pq+qr+rp=0\)
• B. \(p^2+q^2+r^2=2pqr\)
• C. \(p^3+q^3+r^3=3pqr\)
• D. \(p^4+q^4+r^4=4pqr\)

Hint:

For three lines to be concurrent, write the determinant of their coefficients and constant terms, then put it equal to zero.

Answer 1

The Correct Option is C

Concept:
Three straight lines: \[ a_1x+b_1y+c_1=0 \] \[ a_2x+b_2y+c_2=0 \] \[ a_3x+b_3y+c_3=0 \] are concurrent if they pass through a common point. The condition for concurrency is: \[ \begin{vmatrix} a_1 & b_1 & c_1 a_2 & b_2 & c_2 a_3 & b_3 & c_3 \end{vmatrix}=0 \]

Step 1: Write the coefficient matrix.
Given lines are: \[ px+qy+r=0 \] \[ qx+ry+p=0 \] \[ rx+py+q=0 \] So the coefficient matrix is: \[ \begin{vmatrix} p & q & r q & r & p r & p & q \end{vmatrix} \] For the three lines to be concurrent: \[ \begin{vmatrix} p & q & r q & r & p r & p & q \end{vmatrix}=0 \]

Step 2: Expand the determinant.
Expanding along the first row: \[ D=p(rq-p^2)-q(q^2-pr)+r(qp-r^2) \] Now simplify each part: \[ D=pqr-p^3-q^3+pqr+pqr-r^3 \] \[ D=3pqr-(p^3+q^3+r^3) \]

Step 3: Apply concurrency condition.
For concurrency: \[ D=0 \] So: \[ 3pqr-(p^3+q^3+r^3)=0 \] \[ p^3+q^3+r^3=3pqr \]

Step 4: Check the options.
Option (A) \(pq+qr+rp=0\) is not obtained from the determinant.
Option (B) \(p^2+q^2+r^2=2pqr\) is not the concurrency condition.
Option (C) \(p^3+q^3+r^3=3pqr\) is correct.
Option (D) is not related to the determinant expansion. Hence, the correct answer is: \[ \boxed{(C)\ p^3+q^3+r^3=3pqr} \]