Question Number 115112 by bemath last updated on 23/Sep/20
$${What}\:{is}\:{minimum}\:{distance}\:{between}\: \\ $$$${xy}\:=\:\mathrm{4}\:{and}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:=\:\mathrm{4}\:?\: \\ $$$$ \\ $$
Answered by bobhans last updated on 28/Sep/20
$${you}\:{want}\:{a}\:{line}\:{that}'{s}\:{normal}\:{to}\:{both} \\ $$$${curves}\:{and}\:{namely}\:{y}={x} \\ $$$${The}\:{line}\:{intersect}\:{the}\:{hyperbola}\:{at} \\ $$$${x}={y}=\pm\mathrm{2}\:{and}\:{the}\:{circle}\:{at}\:{x}={y}=\pm\sqrt{\mathrm{2}} \\ $$$${We}\:{just}\:{need}\:{to}\:{compute}\:{the}\:{distance} \\ $$$${between}\:\left(\mathrm{2},\mathrm{2}\right)\:{and}\:\left(\sqrt{\mathrm{2}}\:,\sqrt{\mathrm{2}}\:\right). \\ $$$${Minimun}\:{distance}\:=\:\sqrt{\left(\mathrm{2}−\sqrt{\mathrm{2}}\right)^{\mathrm{2}} +\left(\mathrm{2}−\sqrt{\mathrm{2}}\right)^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\sqrt{\mathrm{2}\left(\mathrm{6}−\mathrm{4}\sqrt{\mathrm{2}}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2}\sqrt{\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}}\:=\:\mathrm{2}\left(\sqrt{\mathrm{2}}−\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2}\sqrt{\mathrm{2}}\:−\mathrm{2}\: \\ $$