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mayank sharma Grade: 12
        

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.

8 years ago

Answers : (1)

Ramesh V
70 Points
										

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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  


 



8 years ago
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