Question Number 165122 by mr W last updated on 26/Jan/22
Commented by mr W last updated on 26/Jan/22
$${three}\:{vertices}\:{of}\:{the}\:{square}\:{lie}\:{on} \\ $$$${the}\:{parabola}.\:{find}\:{the}\:{locus}\:{of}\:{the} \\ $$$${fourth}\:{vertex}\:{inside}\:{the}\:{parabola}. \\ $$
Commented by aleks041103 last updated on 26/Jan/22
$${does}\:{the}\:{square}\:{have}\:{a}\:{defined}\:{side}\:{length} \\ $$
Commented by mr W last updated on 26/Jan/22
$${no},\:{it}\:{mustn}'{t}\:{have}\:{fixed}\:{side}\:{length}. \\ $$
Answered by aleks041103 last updated on 26/Jan/22
$${The}\:{curve}\:{is}\:{the}\:{central}\:{branch}\:{of} \\ $$$$\mathrm{8}+\mathrm{92}{X}^{\mathrm{2}} −\mathrm{18}{X}^{\mathrm{4}} +{X}^{\mathrm{6}} −\mathrm{28}{Y}+\mathrm{116}{X}^{\mathrm{2}} {Y}−\mathrm{11}{X}^{\mathrm{4}} {Y}+\mathrm{30}{Y}^{\mathrm{2}} +\mathrm{19}{X}^{\mathrm{2}} {Y}^{\mathrm{2}} −\mathrm{9}{Y}^{\mathrm{3}} =\mathrm{0} \\ $$$${where}\:{X}=\frac{{x}}{{a}}\:{and}\:{Y}=\frac{{y}}{{a}} \\ $$
Commented by aleks041103 last updated on 26/Jan/22
Commented by aleks041103 last updated on 26/Jan/22
$${No}\:{matter}\:{what}\:{I}\:{do},\:{I}\:{can}'{t}\:{separate} \\ $$$${the}\:{three}\:{branches}... \\ $$
Commented by aleks041103 last updated on 26/Jan/22
Commented by mr W last updated on 27/Jan/22
$${great}\:{work}\:{sir}! \\ $$$${i}\:{think}\:{the}\:{complete}\:{locus}\:{has}\:{indeed} \\ $$$${three}\:{branches},\:{since}\:{the}\:{fourth} \\ $$$${vertex}\:{can}\:{be}\:{inside}\:{of}\:{the}\:{parabola} \\ $$$${as}\:{well}\:{as}\:{outside}\:{of}\:{the}\:{parabola}. \\ $$
Commented by mr W last updated on 27/Jan/22
Commented by mr W last updated on 27/Jan/22
$${for}\:{both}\:{cases}\:{there}\:{is}\:{a}\:{minimum} \\ $$$${square}\:\left({blue}\:{ones}\right). \\ $$
Commented by mr W last updated on 27/Jan/22
