Three lines $p x + q y + r = 0$ , $q x + r y + p = 0$ and $r x + p y + q = 0$ are concurrent if

p + q + r = 0

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p^{2} + q^{2} + r^{2} = p q + q r + r p

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

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None of these

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Solution

The correct options are
A $p + q + r = 0$
B $p^{3} + q^{3} + r^{3} = 3 p q r$
C $p^{2} + q^{2} + r^{2} = p q + q r + r p$
Lines are concurrent if $∣ ∣ ∣ ∣ \begin{matrix} p & q & r q & r & p r & p & q \end{matrix} ∣ ∣ ∣ ∣ = 0$
or if $(p + q + r) (p^{2} + q^{2} + r^{2} - p q - q r - r p) = 0$
or if $p^{3} + q^{3} + r^{3} - 3 p q r = 0$

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Similar questions

Three lines $p x + q y + r = 0, q x + r y + p = 0$ and $r x + p y + q = 0$ are concurrent, if

p x + q y + r = 0, q x + r y + p = 0 a n d r x + p y + q = 0

(p - q) x + (q - r) y + (r - p) = 0

(q - r) x + (r - p) y + (p - q) = 0

(r - p) x + (p - q) y + (q - r) = 0

p x + q y + r = 0, q x + r y + p = 0

r x + p y + q = 0

p + q + r

p, q, r

p³+ q³ + r³ – 3pqr = ?

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