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

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

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

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

None of these

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $p^{3} + q^{3} + r^{3} = 3 p q r$
Given lines $p x + q y + r = 0, q x + r y + p = 0$
and $r x + p y + q = 0$ are concurrent.
$∴ ∣ ∣ ∣ ∣ \begin{matrix} p & q & r q & r & p r & p & q \end{matrix} ∣ ∣ ∣ ∣ = 0$
Applying $R_{1} \to R_{1} + R_{2} + R_{3}$ and taking common from $R_{1}$
$(p + q + r) ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 q & r & p r & p & q \end{matrix} ∣ ∣ ∣ ∣ = 0$

$\Rightarrow (p + q + r) (p^{2} + q^{2} + r^{2} - p q - q r - p r) = 0 \Rightarrow p^{3} + q^{3} + r^{3} - 3 p q r = 0$
Therefore, (a) and (c) are the answers.

Suggest Corrections

Similar questions

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

(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

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

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

r x + r y + 1 = 0 (p, q, r

\neq

\frac{p}{p - 1} + \frac{q}{q - 1} + \frac{r}{r - 1} =

Join BYJU'S Learning Program