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

Question

The lines $p x + q y + r = 0, q x + ry + p = 0 an d r x + p y + q = 0$ are concurrent then

p + q + r = 0

p^{3} + q^{3} + r^{3} = 3 pq r

p^{2} + q^{2} + r^{2} - pq - q r - r p = 0

p^{2} + q^{2} + r^{2} = 2 (pq + q r + r p)

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Harper Discussed

The lines

p x + q y + r = 0, q x + ry + p = 0 an d r x + p y + q = 0

are concurrent then

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Text solutionVerified

$∣ ∣ p q r q r p r p q ∣ ∣ = 0$

$p (r q - p^{2}) - q (q^{2} - r p) + r (qp - r^{2}) = 0$

$3 pq r - p^{3} - q^{3} - r^{3} = 0$

$p^{3} + q^{3} + r^{3} = 3 pq r$

$p^{3} + q^{3} + r^{3} - 3 pq r = 0$

$(p + q + r) (p^{2} + q^{2} + r^{2} - p a - q r - p r) = 0$

$\Rightarrow p + q + r = 0$ or

$p^{2} + q^{2} + r^{2} - pq - q r - p r = 0$

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Question Text	The lines $p x + q y + r = 0, q x + ry + p = 0 an d r x + p y + q = 0$ are concurrent then
Answer Type	Text solution:1
Upvotes	150