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

Options

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

Answer

A, B, C

Explanation

For the concurrency of three lines px+gy+r=0.
qx+ry+p=0, rx+py+q =0, we must have
`|{:(p, q, r),(q, r, p), (r, p, q):}|=0`
`"or " 3pqr-p^(3)-q^(3)-r^(3)=0`
`"or " (p+q+r)(p^(2) + q^(2)+r^(2)-pr-pr-rq)= 0`