Question

The three lines $px+qy+r=0,qx+ry+p=0,rx+py+q=0$ are concurrent if

• A. $p+q+r=0$
• B. $p^2+q^2+r^2=pq+qr+pr$
• C. $p^3+q^3+r^3=3pqr$
• D. None of these

Answer 1

The correct option is C

$p^3+q^3+r^3=3pqr$

Explanation for correct option:
Given three lines are $px+qy+r=0,qx+ry+p=0,rx+py+q=0$
The lines are concurrent if determinant is zero
$$\begin{gathered} \Rightarrow \left|\begin{array}{ccc}p& q& r\\ q& r& p\\ r& p& q\end{array}\right|=0 \\ \Rightarrow p(qr-p^2)-q(q^2-pr)+r(pq-r^2)=0 \\ \Rightarrow pqr-p^3-q^3+pqr+pqr-r^3=0 \\ \Rightarrow p^3+q^3+r^3=3pqr \end{gathered}$$
Hence option(C) is correct