Question Number 56914 by rahul 19 last updated on 26/Mar/19
$${The}\:{shortest}\:{distance}\:{between}\:{the}\:{point} \\ $$$$\left(\frac{\mathrm{3}}{\mathrm{2}},\mathrm{0}\right)\:{and}\:{the}\:{curve}\:{y}=\sqrt{{x}}\:,\left({x}>\mathrm{0}\right)\:{is}\:? \\ $$
Answered by Smail last updated on 26/Mar/19
$${d}=\sqrt{\left({x}_{\mathrm{2}} −{x}_{\mathrm{1}} \right)^{\mathrm{2}} +\left({y}_{\mathrm{2}} −{y}_{\mathrm{1}} \right)^{\mathrm{2}} } \\ $$$$=\sqrt{\left({x}−\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} +\left(\sqrt{{x}}−\mathrm{0}\right)^{\mathrm{2}} }=\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\frac{\mathrm{9}}{\mathrm{4}}} \\ $$$${d}'=\frac{\mathrm{2}{x}−\mathrm{2}}{\mathrm{2}\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\frac{\mathrm{9}}{\mathrm{2}}}} \\ $$$${CV}\:{is}\:\:{x}=\mathrm{1}\:\: \\ $$$${d}_{{min}} =\sqrt{\mathrm{1}^{\mathrm{2}} −\mathrm{2}×\mathrm{1}+\frac{\mathrm{9}}{\mathrm{4}}}=\frac{\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$
Commented by rahul 19 last updated on 26/Mar/19
thank you sir!
Answered by tanmay.chaudhury50@gmail.com last updated on 26/Mar/19
$${let}\:\left({x},\sqrt{{x}}\:\right){be}\:{a}\:{point}\:{on}\:{the}\:{curve}. \\ $$$${distance}\:{l}=\sqrt{\left({x}−\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} +\left(\sqrt{{x}}\:−\mathrm{0}\right)^{\mathrm{2}} }\: \\ $$$${l}^{\mathrm{2}} ={x}^{\mathrm{2}} −\mathrm{2}×{x}×\frac{\mathrm{3}}{\mathrm{2}}+\frac{\mathrm{9}}{\mathrm{4}}+{x} \\ $$$${l}^{\mathrm{2}} ={x}^{\mathrm{2}} −\mathrm{2}{x}+\frac{\mathrm{9}}{\mathrm{4}} \\ $$$${l}=\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\frac{\mathrm{9}}{\mathrm{4}}}\: \\ $$$$\frac{{dl}}{{dx}}=\frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\frac{\mathrm{9}}{\mathrm{4}}}}×\left(\mathrm{2}{x}−\mathrm{2}\right) \\ $$$$\frac{{dl}}{{dx}}=\frac{{x}−\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\frac{\mathrm{9}}{\mathrm{4}}}} \\ $$$${for}\:{max}/{min}\:\frac{{dl}}{{dx}}=\mathrm{0}\:\:\:{so}\:{x}=\mathrm{1} \\ $$$$ \\ $$$$\frac{{d}^{\mathrm{2}} {l}}{{dx}^{\mathrm{2}} }=\frac{\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\frac{\mathrm{9}}{\mathrm{4}}\:}\:×\mathrm{1}−\left({x}−\mathrm{1}\right)×\frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\frac{\mathrm{9}}{\mathrm{4}}\:}}×\left(\mathrm{2}{x}−\mathrm{2}\right)}{\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\frac{\mathrm{9}}{\mathrm{4}}\right)} \\ $$$$\left(\frac{{d}^{\mathrm{2}} {l}}{{dx}^{\mathrm{2}} }\right)_{{x}=\mathrm{1}} =\frac{\sqrt{\frac{\mathrm{5}}{\mathrm{4}}}\:×\mathrm{1}}{\left(\frac{\mathrm{5}}{\mathrm{4}}\right)}=\sqrt{\frac{\mathrm{4}}{\mathrm{5}}}\:>\mathrm{0} \\ $$$${so}\:{minimum}\:{distance}\:{is}\:{between}\:{point}\left({x},\sqrt{{x}}\:\right) \\ $$$${that}\:{means}\:\left(\mathrm{1},\mathrm{1}\right)\:{and}\left(\frac{\mathrm{3}}{\mathrm{2}},\mathrm{0}\right)\:{is} \\ $$$${l}=\sqrt{\left(\frac{\mathrm{3}}{\mathrm{2}}−\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{1}−\mathrm{0}\right)^{\mathrm{2}} }\: \\ $$$$=\sqrt{\frac{\mathrm{1}}{\mathrm{4}}+\mathrm{1}}\: \\ $$$$=\sqrt{\frac{\mathrm{5}}{\mathrm{4}}}\: \\ $$$$=\frac{\sqrt{\mathrm{5}}}{\mathrm{2}}\leftarrow{this}\:{is}\:{the}\:{snswer} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Commented by rahul 19 last updated on 26/Mar/19
thank you sir!
Answered by mr W last updated on 26/Mar/19
$${d}=\sqrt{\left({x}−\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} +\left(\sqrt{{x}}−\mathrm{0}\right)^{\mathrm{2}} }=\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\frac{\mathrm{9}}{\mathrm{4}}} \\ $$$$=\sqrt{\left({x}−\mathrm{1}\right)^{\mathrm{2}} +\frac{\mathrm{5}}{\mathrm{4}}}\geqslant\sqrt{\frac{\mathrm{5}}{\mathrm{4}}}=\frac{\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\Rightarrow{d}_{{min}} =\frac{\sqrt{\mathrm{5}}}{\mathrm{2}}\:{at}\:{x}=\mathrm{1} \\ $$
Commented by rahul 19 last updated on 26/Mar/19
thank you sir!