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 \begin{vmatrix}p & q & r \\q &r & p\\r &p &q \end {vmatrix}=0 $
Applying $R_1 \rightarrow R_1+R_2+R_3 $ and taking common from
$R_1 $
$ (p+q+r) \begin {vmatrix}1 & 1 & 1 \\q & r & p \\r & p & q \end {vmatrix}=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.