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Question-81393




Question Number 81393 by ajfour last updated on 12/Feb/20
Commented by ajfour last updated on 12/Feb/20
Q.81331 reposted
$${Q}.\mathrm{81331}\:{reposted} \\ $$
Answered by ajfour last updated on 12/Feb/20
Commented by mr W last updated on 12/Feb/20
Commented by ajfour last updated on 12/Feb/20
or it should be better to maximise  area of △ shown in diagram,  do you get my idea Sir?
$${or}\:{it}\:{should}\:{be}\:{better}\:{to}\:{maximise} \\ $$$${area}\:{of}\:\bigtriangleup\:{shown}\:{in}\:{diagram}, \\ $$$${do}\:{you}\:{get}\:{my}\:{idea}\:{Sir}? \\ $$
Commented by mr W last updated on 12/Feb/20
Commented by mr W last updated on 12/Feb/20
diagram shows the triangle with  maximum area: tangent at P is //  to QR, tangent at Q is // to PR and  tangent at R is // to PQ.
$${diagram}\:{shows}\:{the}\:{triangle}\:{with} \\ $$$${maximum}\:{area}:\:{tangent}\:{at}\:{P}\:{is}\:// \\ $$$${to}\:{QR},\:{tangent}\:{at}\:{Q}\:{is}\://\:{to}\:{PR}\:{and} \\ $$$${tangent}\:{at}\:{R}\:{is}\://\:{to}\:{PQ}. \\ $$
Commented by ajfour last updated on 13/Feb/20
with reference to diagram in  this reposted question, let  G(0,0) , O(−p,0),  D(qcos θ,qsin θ) , E(rcos φ,−rsin φ).  eq. of DE_(−)   y−qsin θ=(((qsin θ+rsin φ)/(qcos θ−rcos φ)))(x−qcos θ)  let y=0, x=l  l=qcos θ−qsin θ(((qcos θ−rcos φ)/(qsin θ+rsin φ)))   = ((qr(sin φcos θ+sin θcos φ))/(qsin θ+rsin φ))  A=(((p+l)(qsin θ+rsin φ))/2)  ⇒ A=((p(qsin θ+rsin φ)+qrsin (θ+φ))/2)  A=((pqsin θ+prsin φ+qr(sin θcos φ+cos θsin φ))/2)  (∂A/∂θ)=0 , (∂A/∂φ)=0  ⇒ pqcos θ+qr(cos θcos φ−sin θsin φ)=0  &  prcos φ+qr(−sin θsin φ+cos θcos φ)=0  ⇒   { ((r(tan θsin φ−cos φ)=p)),((q(tan φsin θ−cos θ)=p)) :}  tan θtan φ=(p/r)sec φ+1=(p/q)sec θ+1  ⇒  rcos φ=qcos θ =x  {otherwise    A=(((p+x)/2))((√(q^2 −x^2 ))+(√(r^2 −x^2 )))  Now   (dA/dx)=0  ⇒  (1/2)((√(q^2 −x^2 ))+(√(r^2 −x^2 )))      = ((x(p+x))/2)[(1/( (√(q^2 −x^2 ))))+(1/( (√(r^2 −x^2 ))))]  (q^2 −x^2 )(r^2 −x^2 )=x^2 (p+x)^2   2px^3 +(p^2 +q^2 +r^2 )x^2 −q^2 r^2 =0  ⇒ (1/x^3 )−(((p^2 +q^2 +r^2 )/(q^2 r^2 )))(1/x)−((2p)/(q^2 r^2 ))=0  D={(p/(q^2 r^2 ))}^2 −(1/(27)){((p^2 +q^2 +r^2 )/(q^2 r^2 ))}^3    D(q^2 r^2 )^3 =p^2 q^2 r^2 −(((p^2 +q^2 +r^2 )/3))^3   Sir, what can be said about the  sign of Discriminant here?    }  or  rsec θ=qsec φ  ⇒ (√(1+sec^2 θ))(√((r^2 /q^2 )sec^2 θ−1))=(p/q)sec θ+1  let  sec θ=t   (1+t^2 )(((r^2 t^2 )/q^2 )−1)=((p^2 t^2 )/q^2 )+((2pt)/q)+1  its a biquadratic...  similarly for tan φ....  ......  AB=2(√((qcos θ+p)^2 +q^2 sin^2 θ))  BC=2(√((qcos θ−rcos φ)^2 +(qsin θ+rsin φ)^2 ))  CA=2(√((rcos φ−p)^2 +r^2 sin^2 φ))  .
$${with}\:{reference}\:{to}\:{diagram}\:{in} \\ $$$${this}\:{reposted}\:{question},\:{let} \\ $$$${G}\left(\mathrm{0},\mathrm{0}\right)\:,\:{O}\left(−{p},\mathrm{0}\right), \\ $$$${D}\left({q}\mathrm{cos}\:\theta,{q}\mathrm{sin}\:\theta\right)\:,\:{E}\left({r}\mathrm{cos}\:\phi,−{r}\mathrm{sin}\:\phi\right). \\ $$$$\underset{−} {{eq}.\:{of}\:{DE}} \\ $$$${y}−{q}\mathrm{sin}\:\theta=\left(\frac{{q}\mathrm{sin}\:\theta+{r}\mathrm{sin}\:\phi}{{q}\mathrm{cos}\:\theta−{r}\mathrm{cos}\:\phi}\right)\left({x}−{q}\mathrm{cos}\:\theta\right) \\ $$$${let}\:{y}=\mathrm{0},\:{x}={l} \\ $$$${l}={q}\mathrm{cos}\:\theta−{q}\mathrm{sin}\:\theta\left(\frac{{q}\mathrm{cos}\:\theta−{r}\mathrm{cos}\:\phi}{{q}\mathrm{sin}\:\theta+{r}\mathrm{sin}\:\phi}\right) \\ $$$$\:=\:\frac{{qr}\left(\mathrm{sin}\:\phi\mathrm{cos}\:\theta+\mathrm{sin}\:\theta\mathrm{cos}\:\phi\right)}{{q}\mathrm{sin}\:\theta+{r}\mathrm{sin}\:\phi} \\ $$$${A}=\frac{\left({p}+{l}\right)\left({q}\mathrm{sin}\:\theta+{r}\mathrm{sin}\:\phi\right)}{\mathrm{2}} \\ $$$$\Rightarrow\:{A}=\frac{{p}\left({q}\mathrm{sin}\:\theta+{r}\mathrm{sin}\:\phi\right)+{qr}\mathrm{sin}\:\left(\theta+\phi\right)}{\mathrm{2}} \\ $$$${A}=\frac{{pq}\mathrm{sin}\:\theta+{pr}\mathrm{sin}\:\phi+{qr}\left(\mathrm{sin}\:\theta\mathrm{cos}\:\phi+\mathrm{cos}\:\theta\mathrm{sin}\:\phi\right)}{\mathrm{2}} \\ $$$$\frac{\partial{A}}{\partial\theta}=\mathrm{0}\:,\:\frac{\partial{A}}{\partial\phi}=\mathrm{0} \\ $$$$\Rightarrow\:{pq}\mathrm{cos}\:\theta+{qr}\left(\mathrm{cos}\:\theta\mathrm{cos}\:\phi−\mathrm{sin}\:\theta\mathrm{sin}\:\phi\right)=\mathrm{0} \\ $$$$\&\:\:{pr}\mathrm{cos}\:\phi+{qr}\left(−\mathrm{sin}\:\theta\mathrm{sin}\:\phi+\mathrm{cos}\:\theta\mathrm{cos}\:\phi\right)=\mathrm{0} \\ $$$$\Rightarrow \\ $$$$\begin{cases}{{r}\left(\mathrm{tan}\:\theta\mathrm{sin}\:\phi−\mathrm{cos}\:\phi\right)={p}}\\{{q}\left(\mathrm{tan}\:\phi\mathrm{sin}\:\theta−\mathrm{cos}\:\theta\right)={p}}\end{cases} \\ $$$$\mathrm{tan}\:\theta\mathrm{tan}\:\phi=\frac{{p}}{{r}}\mathrm{sec}\:\phi+\mathrm{1}=\frac{{p}}{{q}}\mathrm{sec}\:\theta+\mathrm{1} \\ $$$$\Rightarrow\:\:{r}\mathrm{cos}\:\phi={q}\mathrm{cos}\:\theta\:={x} \\ $$$$\left\{{otherwise}\right. \\ $$$$\:\:{A}=\left(\frac{{p}+{x}}{\mathrm{2}}\right)\left(\sqrt{{q}^{\mathrm{2}} −{x}^{\mathrm{2}} }+\sqrt{{r}^{\mathrm{2}} −{x}^{\mathrm{2}} }\right) \\ $$$${Now}\:\:\:\frac{{dA}}{{dx}}=\mathrm{0}\:\:\Rightarrow \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left(\sqrt{{q}^{\mathrm{2}} −{x}^{\mathrm{2}} }+\sqrt{{r}^{\mathrm{2}} −{x}^{\mathrm{2}} }\right) \\ $$$$\:\:\:\:=\:\frac{{x}\left({p}+{x}\right)}{\mathrm{2}}\left[\frac{\mathrm{1}}{\:\sqrt{{q}^{\mathrm{2}} −{x}^{\mathrm{2}} }}+\frac{\mathrm{1}}{\:\sqrt{{r}^{\mathrm{2}} −{x}^{\mathrm{2}} }}\right] \\ $$$$\left({q}^{\mathrm{2}} −{x}^{\mathrm{2}} \right)\left({r}^{\mathrm{2}} −{x}^{\mathrm{2}} \right)={x}^{\mathrm{2}} \left({p}+{x}\right)^{\mathrm{2}} \\ $$$$\mathrm{2}{px}^{\mathrm{3}} +\left({p}^{\mathrm{2}} +{q}^{\mathrm{2}} +{r}^{\mathrm{2}} \right){x}^{\mathrm{2}} −{q}^{\mathrm{2}} {r}^{\mathrm{2}} =\mathrm{0} \\ $$$$\Rightarrow\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }−\left(\frac{{p}^{\mathrm{2}} +{q}^{\mathrm{2}} +{r}^{\mathrm{2}} }{{q}^{\mathrm{2}} {r}^{\mathrm{2}} }\right)\frac{\mathrm{1}}{{x}}−\frac{\mathrm{2}{p}}{{q}^{\mathrm{2}} {r}^{\mathrm{2}} }=\mathrm{0} \\ $$$${D}=\left\{\frac{{p}}{{q}^{\mathrm{2}} {r}^{\mathrm{2}} }\right\}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{27}}\left\{\frac{{p}^{\mathrm{2}} +{q}^{\mathrm{2}} +{r}^{\mathrm{2}} }{{q}^{\mathrm{2}} {r}^{\mathrm{2}} }\right\}^{\mathrm{3}} \\ $$$$\:{D}\left({q}^{\mathrm{2}} {r}^{\mathrm{2}} \right)^{\mathrm{3}} ={p}^{\mathrm{2}} {q}^{\mathrm{2}} {r}^{\mathrm{2}} −\left(\frac{{p}^{\mathrm{2}} +{q}^{\mathrm{2}} +{r}^{\mathrm{2}} }{\mathrm{3}}\right)^{\mathrm{3}} \\ $$$${Sir},\:{what}\:{can}\:{be}\:{said}\:{about}\:{the} \\ $$$${sign}\:{of}\:{Discriminant}\:{here}? \\ $$$$\left.\:\:\right\} \\ $$$${or}\:\:{r}\mathrm{sec}\:\theta={q}\mathrm{sec}\:\phi \\ $$$$\Rightarrow\:\sqrt{\mathrm{1}+\mathrm{sec}\:^{\mathrm{2}} \theta}\sqrt{\frac{{r}^{\mathrm{2}} }{{q}^{\mathrm{2}} }\mathrm{sec}\:^{\mathrm{2}} \theta−\mathrm{1}}=\frac{{p}}{{q}}\mathrm{sec}\:\theta+\mathrm{1} \\ $$$${let}\:\:\mathrm{sec}\:\theta={t} \\ $$$$\:\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\left(\frac{{r}^{\mathrm{2}} {t}^{\mathrm{2}} }{{q}^{\mathrm{2}} }−\mathrm{1}\right)=\frac{{p}^{\mathrm{2}} {t}^{\mathrm{2}} }{{q}^{\mathrm{2}} }+\frac{\mathrm{2}{pt}}{{q}}+\mathrm{1} \\ $$$${its}\:{a}\:{biquadratic}… \\ $$$${similarly}\:{for}\:\mathrm{tan}\:\phi…. \\ $$$$…… \\ $$$${AB}=\mathrm{2}\sqrt{\left({q}\mathrm{cos}\:\theta+{p}\right)^{\mathrm{2}} +{q}^{\mathrm{2}} \mathrm{sin}\:^{\mathrm{2}} \theta} \\ $$$${BC}=\mathrm{2}\sqrt{\left({q}\mathrm{cos}\:\theta−{r}\mathrm{cos}\:\phi\right)^{\mathrm{2}} +\left({q}\mathrm{sin}\:\theta+{r}\mathrm{sin}\:\phi\right)^{\mathrm{2}} } \\ $$$${CA}=\mathrm{2}\sqrt{\left({r}\mathrm{cos}\:\phi−{p}\right)^{\mathrm{2}} +{r}^{\mathrm{2}} \mathrm{sin}\:^{\mathrm{2}} \phi} \\ $$$$. \\ $$
Commented by mr W last updated on 12/Feb/20
ingenious idea!  Δ_(ABC) =4Δ_(shaded)
$${ingenious}\:{idea}! \\ $$$$\Delta_{{ABC}} =\mathrm{4}\Delta_{{shaded}} \\ $$
Commented by mr W last updated on 12/Feb/20
q^2 =p^2 +PQ^2 −2p×PQ cos α  PQ^2 −2pcos α×PQ−(q^2 −p^2 )=0  PQ=p cos α+(√(p^2 cos^2  α+q^2 −p^2 ))  similarly  PR=p cos β+(√(p^2 cos^2  β+r^2 −p^2 ))  A=2Δ_(PQR) =sin (α+β)(p cos α+(√(p^2 cos^2  α+q^2 −p^2 )))(p cos β+(√(p^2 cos^2  β+q^2 −r^2 )))  (∂A/∂α)=[cos (α+β)(p cos α+(√(p^2 cos^2  α+q^2 −p^2 )))−p sin (α+β)sin α(((pcos α+(√(p^2 cos^2  α+q^2 −p^2 )))/( (√(p^2 cos^2  α+q^2 −p^2 )))))](p cos β+(√(p^2 cos^2  β+q^2 −r^2 )))=0  ⇒((sin α tan (α+β))/( (√(((q/p))^2 −sin^2  α))))=1   ...(i)  similarly  ⇒((sin β tan (α+β))/( (√(((r/p))^2 −sin^2  β))))=1   ...(ii)  let (q/p)=δ>1, (r/p)=γ>1  ⇒((sin α)/( (√(δ^2 −sin^2  α))))=(1/(tan (α+β)))=(1/t)  (1+t^2 )sin^2  α=δ^2   ⇒sin α=(δ/( (√(1+t^2 )))) ⇒tan α=(δ/( (√(1+t^2 −δ^2 ))))  similarly  ⇒sin β=(γ/( (√(1+t^2 )))) ⇒tan β=(γ/( (√(1+t^2 −γ^2 ))))  t=tan (α+β)=((tan α+tan β)/(1−tan α tan β))  ⇒t=((δ(√(1+t^2 −γ^2 ))+γ(√(1+t^2 −δ^2 )))/( (√((1+t^2 −δ^2 )(1+t^2 −γ^2 )))−δγ))    c=2PQ=2(p cos α+(√(p^2 cos^2  α+q^2 −p^2 )))  b=2PR=2(p cos β+(√(p^2 cos^2  β+r^2 −p^2 )))  a=2QR=2(PQ×sin α+PR×sin β)
$${q}^{\mathrm{2}} ={p}^{\mathrm{2}} +{PQ}^{\mathrm{2}} −\mathrm{2}{p}×{PQ}\:\mathrm{cos}\:\alpha \\ $$$${PQ}^{\mathrm{2}} −\mathrm{2}{p}\mathrm{cos}\:\alpha×{PQ}−\left({q}^{\mathrm{2}} −{p}^{\mathrm{2}} \right)=\mathrm{0} \\ $$$${PQ}={p}\:\mathrm{cos}\:\alpha+\sqrt{{p}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \:\alpha+{q}^{\mathrm{2}} −{p}^{\mathrm{2}} } \\ $$$${similarly} \\ $$$${PR}={p}\:\mathrm{cos}\:\beta+\sqrt{{p}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \:\beta+{r}^{\mathrm{2}} −{p}^{\mathrm{2}} } \\ $$$${A}=\mathrm{2}\Delta_{{PQR}} =\mathrm{sin}\:\left(\alpha+\beta\right)\left({p}\:\mathrm{cos}\:\alpha+\sqrt{{p}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \:\alpha+{q}^{\mathrm{2}} −{p}^{\mathrm{2}} }\right)\left({p}\:\mathrm{cos}\:\beta+\sqrt{{p}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \:\beta+{q}^{\mathrm{2}} −{r}^{\mathrm{2}} }\right) \\ $$$$\frac{\partial{A}}{\partial\alpha}=\left[\mathrm{cos}\:\left(\alpha+\beta\right)\left({p}\:\mathrm{cos}\:\alpha+\sqrt{{p}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \:\alpha+{q}^{\mathrm{2}} −{p}^{\mathrm{2}} }\right)−{p}\:\mathrm{sin}\:\left(\alpha+\beta\right)\mathrm{sin}\:\alpha\left(\frac{{p}\mathrm{cos}\:\alpha+\sqrt{{p}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \:\alpha+{q}^{\mathrm{2}} −{p}^{\mathrm{2}} }}{\:\sqrt{{p}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \:\alpha+{q}^{\mathrm{2}} −{p}^{\mathrm{2}} }}\right)\right]\left({p}\:\mathrm{cos}\:\beta+\sqrt{{p}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \:\beta+{q}^{\mathrm{2}} −{r}^{\mathrm{2}} }\right)=\mathrm{0} \\ $$$$\Rightarrow\frac{\mathrm{sin}\:\alpha\:\mathrm{tan}\:\left(\alpha+\beta\right)}{\:\sqrt{\left(\frac{{q}}{{p}}\right)^{\mathrm{2}} −\mathrm{sin}^{\mathrm{2}} \:\alpha}}=\mathrm{1}\:\:\:…\left({i}\right) \\ $$$${similarly} \\ $$$$\Rightarrow\frac{\mathrm{sin}\:\beta\:\mathrm{tan}\:\left(\alpha+\beta\right)}{\:\sqrt{\left(\frac{{r}}{{p}}\right)^{\mathrm{2}} −\mathrm{sin}^{\mathrm{2}} \:\beta}}=\mathrm{1}\:\:\:…\left({ii}\right) \\ $$$${let}\:\frac{{q}}{{p}}=\delta>\mathrm{1},\:\frac{{r}}{{p}}=\gamma>\mathrm{1} \\ $$$$\Rightarrow\frac{\mathrm{sin}\:\alpha}{\:\sqrt{\delta^{\mathrm{2}} −\mathrm{sin}^{\mathrm{2}} \:\alpha}}=\frac{\mathrm{1}}{\mathrm{tan}\:\left(\alpha+\beta\right)}=\frac{\mathrm{1}}{{t}} \\ $$$$\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\mathrm{sin}^{\mathrm{2}} \:\alpha=\delta^{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{sin}\:\alpha=\frac{\delta}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:\Rightarrow\mathrm{tan}\:\alpha=\frac{\delta}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} −\delta^{\mathrm{2}} }} \\ $$$${similarly} \\ $$$$\Rightarrow\mathrm{sin}\:\beta=\frac{\gamma}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:\Rightarrow\mathrm{tan}\:\beta=\frac{\gamma}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} −\gamma^{\mathrm{2}} }} \\ $$$${t}=\mathrm{tan}\:\left(\alpha+\beta\right)=\frac{\mathrm{tan}\:\alpha+\mathrm{tan}\:\beta}{\mathrm{1}−\mathrm{tan}\:\alpha\:\mathrm{tan}\:\beta} \\ $$$$\Rightarrow{t}=\frac{\delta\sqrt{\mathrm{1}+{t}^{\mathrm{2}} −\gamma^{\mathrm{2}} }+\gamma\sqrt{\mathrm{1}+{t}^{\mathrm{2}} −\delta^{\mathrm{2}} }}{\:\sqrt{\left(\mathrm{1}+{t}^{\mathrm{2}} −\delta^{\mathrm{2}} \right)\left(\mathrm{1}+{t}^{\mathrm{2}} −\gamma^{\mathrm{2}} \right)}−\delta\gamma} \\ $$$$ \\ $$$${c}=\mathrm{2}{PQ}=\mathrm{2}\left({p}\:\mathrm{cos}\:\alpha+\sqrt{{p}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \:\alpha+{q}^{\mathrm{2}} −{p}^{\mathrm{2}} }\right) \\ $$$${b}=\mathrm{2}{PR}=\mathrm{2}\left({p}\:\mathrm{cos}\:\beta+\sqrt{{p}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \:\beta+{r}^{\mathrm{2}} −{p}^{\mathrm{2}} }\right) \\ $$$${a}=\mathrm{2}{QR}=\mathrm{2}\left({PQ}×\mathrm{sin}\:\alpha+{PR}×\mathrm{sin}\:\beta\right) \\ $$
Commented by mr W last updated on 12/Feb/20
then we can apply the result from  Q61861.
$${then}\:{we}\:{can}\:{apply}\:{the}\:{result}\:{from} \\ $$$${Q}\mathrm{61861}. \\ $$
Commented by ajfour last updated on 12/Feb/20
yes Sir, i viewed it, thanks,  twas good reviewing it..
$${yes}\:{Sir},\:{i}\:{viewed}\:{it},\:{thanks}, \\ $$$${twas}\:{good}\:{reviewing}\:{it}.. \\ $$
Commented by mr W last updated on 13/Feb/20
you have solved the question perfectly  sir! we get a cubic final equation which  can be solved.
$${you}\:{have}\:{solved}\:{the}\:{question}\:{perfectly} \\ $$$${sir}!\:{we}\:{get}\:{a}\:{cubic}\:{final}\:{equation}\:{which} \\ $$$${can}\:{be}\:{solved}. \\ $$
Answered by ajfour last updated on 12/Feb/20
h^2 +k^2 =p^2   (h−(b−a))^2 +(k−c)^2 =r^2    ..(i)  (h−(b+a))^2 +(k−c)^2 =q^2    ..(ii)  subtracting  4ah=r^2 −q^2 +4ab     ....(I)  Rewriting (i)&(ii)  ⇒ −2h(b−a)+(b−a)^2         −2kc+c^2  =r^2 −p^2        ....(iA)  &    −2h(b+a)+(b+a)^2         −2kc+c^2 =q^2 −p^2         ...(iiA)  adding  −4bh−4kc+2c^2 +2(a^2 +b^2 )                       =q^2 +r^2 −2p^2      k=((4bh−2c^2 −2(a^2 +b^2 )+q^2 +r^2 −2p^2 )/(4c))   k =((b{r^2 −q^2 +4ab}−2ac^2 −2a(a^2 +b^2 )+a{q^2 +r^2 −2p^2 })/(4ac))  Now   h^2 +k^2 =p^2   we get a,b,c related.  say  b=f(a,c)  A=4ac  (∂A/∂a)=0 ,  (∂A/∂c)=0     should get us   a, c, f(a,c)=b  Now  AB=(√((2b+2a)^2 +4c^2 ))      BC=4a , CA=(√((2b−2a)^2 +4c^2 ))  .......
$${h}^{\mathrm{2}} +{k}^{\mathrm{2}} ={p}^{\mathrm{2}} \\ $$$$\left({h}−\left({b}−{a}\right)\right)^{\mathrm{2}} +\left({k}−{c}\right)^{\mathrm{2}} ={r}^{\mathrm{2}} \:\:\:..\left({i}\right) \\ $$$$\left({h}−\left({b}+{a}\right)\right)^{\mathrm{2}} +\left({k}−{c}\right)^{\mathrm{2}} ={q}^{\mathrm{2}} \:\:\:..\left({ii}\right) \\ $$$${subtracting} \\ $$$$\mathrm{4}{ah}={r}^{\mathrm{2}} −{q}^{\mathrm{2}} +\mathrm{4}{ab}\:\:\:\:\:….\left({I}\right) \\ $$$${Rewriting}\:\left({i}\right)\&\left({ii}\right) \\ $$$$\Rightarrow\:−\mathrm{2}{h}\left({b}−{a}\right)+\left({b}−{a}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:−\mathrm{2}{kc}+{c}^{\mathrm{2}} \:={r}^{\mathrm{2}} −{p}^{\mathrm{2}} \:\:\:\:\:\:\:….\left({iA}\right) \\ $$$$\&\:\:\:\:−\mathrm{2}{h}\left({b}+{a}\right)+\left({b}+{a}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:−\mathrm{2}{kc}+{c}^{\mathrm{2}} ={q}^{\mathrm{2}} −{p}^{\mathrm{2}} \:\:\:\:\:\:\:\:…\left({iiA}\right) \\ $$$${adding} \\ $$$$−\mathrm{4}{bh}−\mathrm{4}{kc}+\mathrm{2}{c}^{\mathrm{2}} +\mathrm{2}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={q}^{\mathrm{2}} +{r}^{\mathrm{2}} −\mathrm{2}{p}^{\mathrm{2}} \\ $$$$\:\:\:{k}=\frac{\mathrm{4}{bh}−\mathrm{2}{c}^{\mathrm{2}} −\mathrm{2}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)+{q}^{\mathrm{2}} +{r}^{\mathrm{2}} −\mathrm{2}{p}^{\mathrm{2}} }{\mathrm{4}{c}} \\ $$$$\:{k}\:=\frac{{b}\left\{{r}^{\mathrm{2}} −{q}^{\mathrm{2}} +\mathrm{4}{ab}\right\}−\mathrm{2}{ac}^{\mathrm{2}} −\mathrm{2}{a}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)+{a}\left\{{q}^{\mathrm{2}} +{r}^{\mathrm{2}} −\mathrm{2}{p}^{\mathrm{2}} \right\}}{\mathrm{4}{ac}} \\ $$$${Now}\:\:\:{h}^{\mathrm{2}} +{k}^{\mathrm{2}} ={p}^{\mathrm{2}} \\ $$$${we}\:{get}\:{a},{b},{c}\:{related}. \\ $$$${say}\:\:{b}={f}\left({a},{c}\right) \\ $$$${A}=\mathrm{4}{ac} \\ $$$$\frac{\partial{A}}{\partial{a}}=\mathrm{0}\:,\:\:\frac{\partial{A}}{\partial{c}}=\mathrm{0}\:\:\: \\ $$$${should}\:{get}\:{us}\:\:\:{a},\:{c},\:{f}\left({a},{c}\right)={b} \\ $$$${Now}\:\:{AB}=\sqrt{\left(\mathrm{2}{b}+\mathrm{2}{a}\right)^{\mathrm{2}} +\mathrm{4}{c}^{\mathrm{2}} } \\ $$$$\:\:\:\:{BC}=\mathrm{4}{a}\:,\:{CA}=\sqrt{\left(\mathrm{2}{b}−\mathrm{2}{a}\right)^{\mathrm{2}} +\mathrm{4}{c}^{\mathrm{2}} } \\ $$$$……. \\ $$
Commented by mr W last updated on 12/Feb/20
it′s great sir! thanks!    we get a relation (condition) for a,b,c,  say F(a,b,c)=0  to get maximum of A=4ac, we can  use P=4ac+λF(a,b,c) and  (∂P/∂a)=4c+λ((∂F(a,b,c))/∂a)=0  (∂P/∂b)=λ((∂F(a,b,c))/∂b)=0  (∂P/∂c)=4a+λ((∂F(a,b,c))/∂c)=0  (∂P/∂λ)=F(a,b,c)=0    ⇒F(a,b,c)=0    ...(I)  ⇒((∂F(a,b,c))/∂b)=0  ...(II)  ⇒a((∂F(a,b,c))/∂a)=c((∂F(a,b,c))/∂c)   ...(III)  we get a,b,c from these three eqn.
$${it}'{s}\:{great}\:{sir}!\:{thanks}! \\ $$$$ \\ $$$${we}\:{get}\:{a}\:{relation}\:\left({condition}\right)\:{for}\:{a},{b},{c}, \\ $$$${say}\:{F}\left({a},{b},{c}\right)=\mathrm{0} \\ $$$${to}\:{get}\:{maximum}\:{of}\:{A}=\mathrm{4}{ac},\:{we}\:{can} \\ $$$${use}\:{P}=\mathrm{4}{ac}+\lambda{F}\left({a},{b},{c}\right)\:{and} \\ $$$$\frac{\partial{P}}{\partial{a}}=\mathrm{4}{c}+\lambda\frac{\partial{F}\left({a},{b},{c}\right)}{\partial{a}}=\mathrm{0} \\ $$$$\frac{\partial{P}}{\partial{b}}=\lambda\frac{\partial{F}\left({a},{b},{c}\right)}{\partial{b}}=\mathrm{0} \\ $$$$\frac{\partial{P}}{\partial{c}}=\mathrm{4}{a}+\lambda\frac{\partial{F}\left({a},{b},{c}\right)}{\partial{c}}=\mathrm{0} \\ $$$$\frac{\partial{P}}{\partial\lambda}={F}\left({a},{b},{c}\right)=\mathrm{0} \\ $$$$ \\ $$$$\Rightarrow{F}\left({a},{b},{c}\right)=\mathrm{0}\:\:\:\:…\left({I}\right) \\ $$$$\Rightarrow\frac{\partial{F}\left({a},{b},{c}\right)}{\partial{b}}=\mathrm{0}\:\:…\left({II}\right) \\ $$$$\Rightarrow{a}\frac{\partial{F}\left({a},{b},{c}\right)}{\partial{a}}={c}\frac{\partial{F}\left({a},{b},{c}\right)}{\partial{c}}\:\:\:…\left({III}\right) \\ $$$${we}\:{get}\:{a},{b},{c}\:{from}\:{these}\:{three}\:{eqn}. \\ $$

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