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