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Question Number 63438 by ajfour last updated on 04/Jul/19

Commented by ajfour last updated on 04/Jul/19

Determine a:b:c if the inscribed △ABC has constant perimeter but maximum area.

Determinea:b:ciftheinscribed△ABChasconstantperimeterbutmaximumarea.

Commented by ajfour last updated on 04/Jul/19

MjS Sir, mrW sir, Tanmay Sir, please try..

MjSSir,mrWsir,TanmaySir,pleasetry..

Answered by ajfour last updated on 04/Jul/19

A(h,h^2 ) B(h+ccos θ, h^2 +csin θ) C(h+bcos φ, h^2 +bsin φ) a+b+c=p ......(i) h^2 +csin θ=(h+ccos θ)^2 ...(ii) ⇒ sin θ=ccos^2 θ+2hcos θ ⇒ ccos θ=tan θ−2h h^2 +bsin φ=(h+bcos φ)^2 ....(iii) ⇒ bcos φ=tan φ−2h (ccos θ−bcos φ)^2 +(csin θ−bsin φ)^2 =a^2 .....(iv) ⇒ b^2 +c^2 −a^2 =2bccos (φ−θ) ⇒ −p^2 −2(b+c)p=2bccos (φ−θ) −p^2 −2p[((tan θ−2h)/(cos θ))+((tan φ−2h)/(cos φ))] = 2(tan θ−2h)(tan φ−2h)(1+tan θtan φ) △=((cd)/2) d=b(sin φ−cos φtan θ)cos θ ⇒ △=((bc)/2)(bsin φ−bcos φtan θ)cos θ △=((bc)/2)sin (φ−θ) 2△=(((tan θ−2h)(tan φ−2h)sin (φ−θ))/(cos θcos φ)) 2△=(tan θ−2h)(tan φ−2h)(tan φ−tan θ) −p^2 −2p[((tan θ−2h)/(cos θ))+((tan φ−2h)/(cos φ))] = 2(tan θ−2h)(tan φ−2h)(1+tan θtan φ) Now let tan θ=t, tan φ=s ⇒ −p^2 −2p[(t−2h)(√(1+t^2 ))+(s−2h)(√(1+s^2 )) ] =2(t−2h)(s−2h)(1+st) but i′m afraid its not going to yield!

A(h,h2)B(h+ccosθ,h2+csinθ)C(h+bcosϕ,h2+bsinϕ)a+b+c=p......(i)h2+csinθ=(h+ccosθ)2...(ii)⇒sinθ=ccos2θ+2hcosθ⇒ccosθ=tanθ−2hh2+bsinϕ=(h+bcosϕ)2....(iii)⇒bcosϕ=tanϕ−2h(ccosθ−bcosϕ)2+(csinθ−bsinϕ)2=a2.....(iv)⇒b2+c2−a2=2bccos(ϕ−θ)⇒−p2−2(b+c)p=2bccos(ϕ−θ)−p2−2p[tanθ−2hcosθ+tanϕ−2hcosϕ]=2(tanθ−2h)(tanϕ−2h)(1+tanθtanϕ)△=cd2d=b(sinϕ−cosϕtanθ)cosθ⇒△=bc2(bsinϕ−bcosϕtanθ)cosθ△=bc2sin(ϕ−θ)2△=(tanθ−2h)(tanϕ−2h)sin(ϕ−θ)cosθcosϕ2△=(tanθ−2h)(tanϕ−2h)(tanϕ−tanθ)−p2−2p[tanθ−2hcosθ+tanϕ−2hcosϕ]=2(tanθ−2h)(tanϕ−2h)(1+tanθtanϕ)Nowlettanθ=t,tanϕ=s⇒−p2−2p[(t−2h)1+t2+(s−2h)1+s2]=2(t−2h)(s−2h)(1+st)buti′mafraiditsnotgoingtoyield!

Commented by ajfour last updated on 04/Jul/19

−p^2 −2p[((tan θ−2h)/(cos θ))+((tan φ−2h)/(cos φ))] = 2(tan θ−2h)(tan φ−2h)(1+tan θtan φ) ........(I) 2△=(tan θ−2h)(tan φ−2h)(tan φ−tan θ) ........(II) To find θ,φ such that △ is max.

−p2−2p[tanθ−2hcosθ+tanϕ−2hcosϕ]=2(tanθ−2h)(tanϕ−2h)(1+tanθtanϕ)........(I)2△=(tanθ−2h)(tanϕ−2h)(tanϕ−tanθ)........(II)Tofindθ,ϕsuchthat△ismax.

Commented by mr W last updated on 04/Jul/19

a hard question sir...

ahardquestionsir...

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