If lines $p x + q y + r = 0, q x + r y + p = 0$ and $r x + p y + q = 0$ are concurrent, then the value of $p + q + r$ is: (where $p, q, r$ are distinct).

0

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Solution

The correct option is A $0$
For concurrency of three lines
$p x + q y + r = 0; q x + r y + p = 0; r x + p y + q = 0$
We must have, $∣ ∣ ∣ ∣ \begin{matrix} p & q & r q & r & p r & p & q \end{matrix} ∣ ∣ ∣ ∣ = 0$
On expanding we have,
$3 p q r - p^{3} - q^{3} - r^{3} = 0 \Rightarrow (p + q + r) (p^{2} + q^{2} + r^{2} - p q - p r - r q) = 0$
$\Rightarrow (p + q + r) \frac{1}{2} ((p - q)^{2} + (q - r)^{2} + (r - p)^{2})$
$\Rightarrow p + q + r = 0 (∵ p, q, r are distinct)$

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