If lines `p x+q y+r=0,q x+r y+p=0 and r x+p y+q=0` are concurrent, then prove that `p+q+r=0(where, p ,q ,r` are distinct`)dot`

Options

A. `p+q+r=0`
B. `p^(2)+q^(2)+r^(2)=pq+qr+rp`
C. `p^(3)+q^(3)+r^(3)=3pqr`
D. none of these

Answer

C

Explanation

The lines `px+qy+r=0, qx+ry+p=0` and `rx+py+q=0` are concurrent, if
`|{:(p,q,r),(q,r,p),(r,p,q):}|=0`
`implies (p+q+r)^(2)(p^(2)+q^(2)+r^(2)-pq-qr-rp)=0`
`implies p^(3)+q^(3)+r^(3)-3pqr=0implies p^(3)+q^(3)=3pqr`