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What is the smallest distnace between the origin and a point on the graph of \(y = \frac{1}{\sqrt{2}} (x^2 - 18)\)?

 Feb 6, 2022
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What is the smallest distance between the origin and a point on the graph?

 

Hello Guest!

 

\(\color{blue}f(x)= y=\dfrac{1}{\sqrt{2}} (x^2 - 18)\\ y= \dfrac{x^2}{\sqrt{2}}-9\cdot \sqrt{2}\\ \color{blue}\dfrac{dy}{dx}=y\ '=\sqrt{2}\cdot x\)

 

 

\(g(x)=-\frac{1}{y\ '}(x-0)+0\\ g(x)=-\frac{1}{\sqrt{2}\cdot x}(x-0)+0\ |\ point\ direction\ equation\\ \color{blue}g(x)= -\frac{1}{\sqrt{2}}\)  

\(\color{blue}g(x)=f(x)\\-\dfrac{1}{\sqrt{2}}=\dfrac{1}{\sqrt{2}} (x^2 - 18)\\ x^2-18+1=0\\ x= \sqrt{17}\)

\(x=4.1231\)

\( y=\dfrac{1}{\sqrt{2}} (x^2 - 18)\\ y=\dfrac{1}{\sqrt{2}} (17 - 18)=-\dfrac{1}{\sqrt{2}}\\\)

\(y=0.7071\)

\(d=\sqrt{x^2+y^2}=\sqrt{17+\frac{1}{2}}\)

\(d=4.1833\)


 The smallest distance between the origin and a point on the graph is 4.1833 .

laugh  !

 Feb 6, 2022
edited by asinus  Feb 6, 2022

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