If lines px+qy+r=0,qx+ry+p=0 and rx+py+q=0 are concurrent, then the value of p+q+r is: (where p,q,r are distinct).
Open in App
Solution
For concurrency of three lines px+qy+r=0;qx+ry+p=0;rx+py+q=0
We must have, ∣∣
∣∣pqrqrprpq∣∣
∣∣=0
On expanding we have, 3pqr−p3−q3−r3=0⇒(p+q+r)(p2+q2+r2−pq−pr−rq)=0 ⇒(p+q+r)12((p−q)2+(q−r)2+(r−p)2) ⇒p+q+r=0(∵p,q,r are distinct)