Sir please solve every step for this question 

Asked by Surjitsp6u | 3rd Sep, 2019, 10:48: PM

Expert Answer:

Let the equations of n straight lines be
y=mix+ci for i=1,2,3,...n
Let O be the origin and R(h,k) as the locus of R
Equation of the line OR is y=mx
i.e. k-mh=0
N o w comma space O R equals square root of h squared plus k squared end root
R subscript i equals open parentheses fraction numerator c subscript i over denominator m minus m subscript i end fraction comma m cross times fraction numerator c subscript i over denominator m minus m subscript i end fraction close parentheses
O R subscript i equals square root of open parentheses fraction numerator c subscript i over denominator m minus m subscript i end fraction close parentheses squared plus open parentheses fraction numerator m c subscript i over denominator m minus m subscript i end fraction close parentheses squared end root equals open parentheses fraction numerator c subscript i over denominator m minus m subscript i end fraction close parentheses square root of 1 plus m squared end root
G i v e n space t h a t colon
fraction numerator n over denominator O R end fraction equals fraction numerator 1 over denominator O R subscript 1 end fraction plus fraction numerator 1 over denominator O R subscript 2 end fraction plus... plus fraction numerator 1 over denominator O R subscript n end fraction
fraction numerator n over denominator square root of h squared plus k squared end root end fraction equals fraction numerator m minus m subscript 1 over denominator c subscript 1 square root of 1 plus m squared end root end fraction plus fraction numerator m minus m subscript 2 over denominator c subscript 2 square root of 1 plus m squared end root end fraction... plus fraction numerator m minus m subscript n over denominator c subscript n square root of 1 plus m squared end root end fraction
S u b s t i t u t i n g space k equals m h
rightwards double arrow fraction numerator n over denominator h square root of 1 plus m squared end root end fraction equals fraction numerator begin display style k over h end style minus m subscript 1 over denominator c subscript 1 square root of 1 plus m squared end root end fraction plus fraction numerator begin display style k over h end style minus m subscript 2 over denominator c subscript 2 square root of 1 plus m squared end root end fraction... plus fraction numerator begin display style k over h end style minus m subscript n over denominator c subscript n square root of 1 plus m squared end root end fraction
rightwards double arrow n over h equals fraction numerator begin display style k over h minus m subscript 1 end style over denominator c subscript 1 end fraction plus fraction numerator begin display style k over h minus m subscript 2 end style over denominator c subscript 2 end fraction... plus fraction numerator begin display style k over h minus m subscript n end style over denominator c subscript n end fraction
rightwards double arrow n over h equals k over h open parentheses 1 over c subscript 1 plus 1 over c subscript 2 plus.... plus 1 over c subscript n close parentheses minus open parentheses fraction numerator begin display style m subscript 1 end style over denominator c subscript 1 end fraction plus fraction numerator begin display style m subscript 2 end style over denominator c subscript 2 end fraction... plus fraction numerator begin display style m subscript n end style over denominator c subscript n end fraction close parentheses
T a k e space 1 over c subscript 1 plus 1 over c subscript 2 plus.... plus 1 over c subscript n equals A space a n d space fraction numerator begin display style m subscript 1 end style over denominator c subscript 1 end fraction plus fraction numerator begin display style m subscript 2 end style over denominator c subscript 2 end fraction... plus fraction numerator begin display style m subscript n end style over denominator c subscript n end fraction equals B
rightwards double arrow n over h equals k over h A minus B
rightwards double arrow n equals k A minus h B
H e n c e comma space t h e space l o c u s space o f space R space i s space B X minus A y plus n equals 0 space w h i c h space i s space a space s t r a i g h t space l i n e

Answered by Renu Varma | 19th Sep, 2019, 10:53: AM

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