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Question Number 169669 by ajfour last updated on 05/May/22

Answered by mr W last updated on 06/May/22

Commented by mr W last updated on 06/May/22

P(p,q) ((a−q)/p)=(a/b) ⇒q=a(1−(p/b)) eqn. of BM: (x/p)+(y/a)=1 eqn. of AN: (x/b)+(y/q)=1 ax_Q +py_Q =ap qx_Q +by_Q =bq ⇒x_Q =((bp(a−q))/(ab−pq)) ⇒y_Q =((aq(b−p))/(ab−pq)) PQ^2 =[p−((bp(a−q))/(ab−pq))]^2 +[q−((aq(b−p))/(ab−pq))]^2 PQ^2 =(([(b−p)^2 +(a−q)^2 ]p^2 q^2 )/((ab−pq)^2 )) PQ^2 =(((a^2 +b^2 +p^2 +q^2 −2bp−2aq)p^2 q^2 )/((ab−pq)^2 )) let λ=(a/b), ξ=(p/b), μ=(q/a)=1−ξ ((PQ^2 )/b^2 )=(((λ^2 +1+ξ^2 +λ^2 μ^2 −2ξ−2λ^2 μ)ξ^2 μ^2 )/((1−ξμ)^2 )) ((PQ^2 )/b^2 )=(((λ^2 +1+ξ^2 +λ^2 (1−ξ)^2 −2ξ−2λ^2 (1−ξ))ξ^2 (1−ξ)^2 )/((1−ξ(1−ξ))^2 )) ((PQ^2 )/b^2 )=(([(λ^2 +1)ξ^2 −2ξ+1]ξ^2 (1−ξ)^2 )/((ξ^2 −ξ+1)^2 )) ((PQ)/b)=Φ=((ξ(1−ξ)(√((λ^2 +1)ξ^2 −2ξ+1)))/(ξ^2 −ξ+1)) such that PQ is maximum, (dΦ/dξ)=0 example: λ=(a/b)=0.5 ⇒ξ=0.387, Φ=0.1999 ⇒PQ_(max) =0.1999b example: λ=(a/b)=1 ⇒ξ=0.5, Φ=0.2357 ⇒PQ_(max) =0.2357b

P(p,q)a−qp=ab⇒q=a(1−pb)eqn.ofBM:xp+ya=1eqn.ofAN:xb+yq=1axQ+pyQ=apqxQ+byQ=bq⇒xQ=bp(a−q)ab−pq⇒yQ=aq(b−p)ab−pqPQ2=[p−bp(a−q)ab−pq]2+[q−aq(b−p)ab−pq]2PQ2=[(b−p)2+(a−q)2]p2q2(ab−pq)2PQ2=(a2+b2+p2+q2−2bp−2aq)p2q2(ab−pq)2letλ=ab,ξ=pb,μ=qa=1−ξPQ2b2=(λ2+1+ξ2+λ2μ2−2ξ−2λ2μ)ξ2μ2(1−ξμ)2PQ2b2=(λ2+1+ξ2+λ2(1−ξ)2−2ξ−2λ2(1−ξ))ξ2(1−ξ)2(1−ξ(1−ξ))2PQ2b2=[(λ2+1)ξ2−2ξ+1]ξ2(1−ξ)2(ξ2−ξ+1)2PQb=Φ=ξ(1−ξ)(λ2+1)ξ2−2ξ+1ξ2−ξ+1suchthatPQismaximum,dΦdξ=0example:λ=ab=0.5⇒ξ=0.387,Φ=0.1999⇒PQmax=0.1999bexample:λ=ab=1⇒ξ=0.5,Φ=0.2357⇒PQmax=0.2357b

Commented by mr W last updated on 06/May/22

Commented by ajfour last updated on 06/May/22

thanks sir, i had expected the question to be an easy one.

thankssir,ihadexpectedthequestiontobeaneasyone.

Commented by Tawa11 last updated on 06/May/22

Great sir.

Greatsir.

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