given n straight lines and a fixed point O.through O a straight line is drawn meeting these points at R1,R2....,Rn and a point R is taken on it such that, n/OR = 1/OR1 + ......+ 1/ORn.show that locus of r is a straight line.

Let the n straight lines be : y=m_ix+c_i  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

R_i = (c_i/(m-m_i),m.c_i/(m-m_i) )

OR_i = { (c_i/(m-m_i)*(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-m_n)/(c_n*(1+m₂)_^1/2)

on substituting k=mh above , we have

n/h = (k/h - m₁)/c₁ +   (k/h - m₂)/c_2+....                                                                          

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                                                         

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Regards

Ramesh