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The lines px+qy+r=0 qx+ry+p=0 rx+py+q=0 are concurrent if a)p+q+r=0b)p^2+q^2+r^2=pq+qr+rpc)p^3+q^3+r^3=3pqrd)none of theseAnswer given is a,b,c

The lines px+qy+r=0 qx+ry+p=0 rx+py+q=0 are concurrent if a)p+q+r=0b)p^2+q^2+r^2=pq+qr+rpc)p^3+q^3+r^3=3pqrd)none of theseAnswer given is a,b,c

Grade:12

1 Answers

Arun
25750 Points
6 years ago
The condition for concurrency of three lines
a1x +b1y+c1 = 0,
a2x+b2y+c2 = 0 and
a3x+b3y+c3=0 is
The determinant

|a1 b1 c1|
|a2 b2 c2|
|a3 b3 c3|

= 0

Hence the condition for given lines to be concurrent is
|p q r|
|q r p|
|r p q|

= 0
=> p(rq-p²) –q(q²-rp) +r(pq-r²) = 0
=> prq - p³ - q³+qrp +rpq - r³ = 0
= > -p³-q³-r³ = -3pqr
=> p³+q³+r³ = 3pqr
=> p³+q³+r³ - 3pqr = 0
=> (p+q+r)(p²+q²+r²-pq-qr-rp) = 0
=> p+q+r = 0 or
=> p²+q²+r² = pq+qr+rp
 
Hence option A, B, C are correct.

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