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```
given n straight lines and a fixed point O.through O a straight line is drawn meeting these points at R 1 ,R 2 ....,R n and a point R is taken on it such that, n/OR = 1/OR 1 + ......+ 1/OR n .show that locus of r is a straight line.

```
11 years ago

```							Let the n straight lines be : y=mix+ci  for i=1,2,3...n
let O be origin and R(h,k) be locus of R
equation of line OR is : y-mx=0
so, k-mh=0
OR= (x2+y2)1/2
Ri = (ci/(m-mi),m.ci/(m-mi) )
ORi = { (ci/(m-mi)*(1+m2)1/2 )
given n/OR = 1/OR1 + 1/OR2 + ....
n/(x2+y2)1/2 = (m-m1)/(c1*(1+m2)1/2) + (m-m2)/(c2*(1+m2)1/2) +............+(m-mn)/(cn*(1+m2)1/2)
on substituting k=mh above , we have
n/h = (k/h - m1)/c1 +   (k/h - m2)/c2+....
n/h = k/h(1/c1 + 1/c2+ .....) - (m1/c1 + m2/c2+ ....)
put A=(1/c1 + 1/c2+ .....)
B=(m1/c1 + m2/c2+ ....)
n/h = A*k/h - B
or n = kA - Bh               this is a straight line
or we can say its locus as Bx-Ay+n=0
--
Regards
Ramesh

```
11 years ago
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