Question

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

• A. $p+q+r=0$
• B. $p^2+q^2+r^2=pr+rq$
  
• C. $p^3+q^3+r^3=3pqr$
  
• D. None of the above
  

Answer 1

Answer: (A), (C)

Given lines $px+qy+r=0,qx+ry+p=0$
and $rx+py+q=0$ are concurrent.
$\therefore \left|\begin{array}{ccc}p& q& r\\ q& r& p\\ r& p& q\end{array}\right|=0$
Applying $R_1\to R_1+R_2+R_3$ and taking common from
$R_1$
$(p+q+r)\left|\begin{array}{ccc}1& 1& 1\\ q& r& p\\ r& p& q\end{array}\right|=0$
$\Rightarrow (p+q+r)(p^2+q^2+r^2-pq-qr-pr)=0$
$\Rightarrow p^3+q^3+r^3-3pqr=0$
Therefore, (a) and (c) are the answers.