Question

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

Answer 1

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Answer 2

Answer:

The given lines are concurrent if $3pqr=p^3+q^3+r^3$.

Step-by-step explanation:

Three straight lines $a_1x+b_1y+c_1=0$, $a_2x+b_2y+c_2=0$ and $a_3x+b_3y+c_3=0$ are concurrent if

$\left[\begin{array}{ccc}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{array}\right]$

The given three lines are

$px+qy+r=0$

$qx+ry+p=0$

$rx+py+q=0$

These three straight lines are concurrent if

$\left[\begin{array}{ccc}p&q&r\\q&r&p\\r&p&q\end{array}\right]$

Expand by row 1.

$p(qr-p^2)-q(q^2-pr)+r(pq-r^2)=0$

$pqr-p^3-q^3+pqr+pqr-r^3=0$

$-p^3-q^3+3pqr-r^3=0$

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

Therefore the given lines are concurrent if $3pqr=p^3+q^3+r^3$.