p,q,r,s in proportion then prove that 

(p-r)q²/(q-s)rs=(p²-q²-pq)/(r²-s²-rd)

 

 

Asked by umeshkumargouda1 | 16th Jun, 2021, 07:15: PM

Expert Answer:

As p, q, r, s are in proportion, we have
 
straight p over straight q equals straight r over straight s equals straight k
rightwards double arrow straight p equals kq space and space straight r equals ks
LHS equals fraction numerator open parentheses straight p minus straight r close parentheses straight q squared over denominator open parentheses straight q minus straight s close parentheses rs end fraction equals fraction numerator open parentheses kq minus ks close parentheses straight q squared over denominator open parentheses straight q minus straight s close parentheses kss end fraction equals fraction numerator straight k open parentheses straight q minus straight s close parentheses straight q squared over denominator open parentheses straight q minus straight s close parentheses ks squared end fraction equals straight q squared over straight s squared
RHS equals fraction numerator straight p squared minus straight q squared minus pq over denominator straight r squared minus straight s squared minus rs end fraction equals fraction numerator straight k squared straight q squared minus straight q squared minus kq squared over denominator straight k squared straight s squared minus straight s squared minus ks squared end fraction equals fraction numerator straight q squared open parentheses straight k squared minus 1 minus straight k close parentheses over denominator straight s squared open parentheses straight k squared minus 1 minus straight k close parentheses end fraction equals straight q squared over straight s squared
Hence comma space fraction numerator open parentheses straight p minus straight r close parentheses straight q squared over denominator open parentheses straight q minus straight s close parentheses rs end fraction equals fraction numerator straight p squared minus straight q squared minus pq over denominator straight r squared minus straight s squared minus rs end fraction

Answered by Renu Varma | 16th Jun, 2021, 10:30: PM